What are some effective ways for someone without a graduate degree to increase their chances of getting hired as a lecturer?

Gain specific professional experience in the subject you want to teach.

In my (engineering) department, all of the teaching staff who do not hold PhDs (or are not current PhD students) have industry experience in the subject they are teaching.

In other words, if you are a professional penetration tester, you may be hired to teach computer security. If you work for a major telecom, you may be hired to teach computer networks. Kickstart a successful electronics startup, and you may be hired to teach circuits. You get the idea.

The point is to have some domain-specific expertise you can bring to the table (because university professors are supposed to be subject matter experts, or at least be able to pretend to be). If it's not from research, professional experience can also work.

Part of the problem with academia right now is that supply (the number of unemployed or underemployed PhDs) greatly outnumbers the number of positions available. One of my colleagues told me he got 400 applications for just one t-t position, this isn't unusual.

Thus, universities don't have to lower their selection criteria in order to find willing and able candidates for even the worst of academic jobs -- the per-course, per-semester adjunct lecturer.¹

Unless you are a relatively Famous Person®, it'll be very difficult to get past even the initial screening for a lectureship without a PhD at a university.

You may have much better luck at community-colleges, technical colleges, online colleges, and polytechnics -- but even there, the economics of oversupply have meant jobs are tight for all.

_{Fn1: There is so much competition in the USA that some adjuncts are paid less than $1500 per course-semester -- that means that even if they taught a 5:5 schedule (10 courses a year), they would still meet the US federal poverty guidelines for food stamps.}

I am going to try to focus on your specific question: ways to increase the chance of getting hired. I'll try to avoid digressing into a discussion of the pros and cons of such a career, but I'd recommend you explore those further - my impression from your questions and comments is that you don't yet have a clear sense of what such a job is like. Also, my answer will be based on the situation in academia in general, based on my knowledge of it: it is always possible that your university is a significant outlier.

Since there is a lot of confusion over job titles across countries, in this post I'll use the word instructor to refer to a person who teaches in a university, and has full responsibility for the classes they teach. It should be distinguished from a teaching assistant whose teaching is done under the supervision of an instructor. In general, the basic qualifications to be hired as an instructor are content expertise and a record of successful university teaching.

Content Expertise

One of the features that most strongly distinguishes university education from high school is that teaching is done by content experts - people whose knowledge of their content area goes well beyond the introductory level. Have you ever had a teacher who you felt was learning their material one chapter ahead of the students? At a university, the goal is to get as far from that situation as possible. The most basic way to demonstrate a level of expertise is to earn a postgraduate degree. At many universities, a masters degree is an absolute minimum requirement for any instructor, and a doctorate is usually preferred.

I think it is easy for people to underestimate the importance of this. After all, if you learned calculus your first year of college, why should you need to take 5-8 years of more advanced classes (many not involving calculus in any obvious way) before you can teach it? But in my personal experience, I think that having deeper experience in mathematics (a doctorate and research) really has helped me understand calculus - what it can be used for, different ways to interpret its results, how it relates to other parts of mathematics - at a much deeper level than I could as a college freshman, or even as a college graduate. And it benefits my teaching: I can help students make connections, discover alternate approaches, etc, in a way that I couldn't do otherwise.

(In particular, it is not likely to be helpful to take the point of view that you don't need a postgraduate degree because it won't help you teach. For one thing, the people hiring you are the university faculty - they will almost all have postgraduate degrees, and won't take kindly to someone suggesting they are of little value.)

It's quite unusual for someone with only a bachelor's degree to be a university instructor, and it would be extremely unusual for someone without a bachelor's degree. I'm rather startled to hear you mention that your university has such people working as instructors. (Is it possible they are actually teaching assistants?) If it's really the case, this may not be a good sign - it may suggest that your university is operating below international academic standards.

Research experience beyond the PhD is viewed as an even stronger sign of content expertise. If you aren't interested in research, then you should know that this is going to work to your detriment when you compete with people who have been active in research.

In some cases, there can be alternative ways to demonstrate an appropriate level of expertise. In some fields, industry or clinical experience can serve as a sign of expertise, even if you don't have a postgraduate degree. But it usually has to be pretty significant - the institution wants to be convinced that your work in the field has been extensive enough to fill in any gaps in your formal education. And part of the benefit of hiring someone with industry experience is that their teaching can be oriented toward techniques, tools, approaches, etc, that are actually widely used in industry - so they will want to be convinced that your experience is broad enough that your sense of industry standards is accurate. (For instance, if you only worked at one company that had some oddball approach, you might have a skewed view that this was common in the industry - if you train your students only in the oddball approach, they won't be well prepared for other industry jobs.) So I'd say they'd want you to have pretty extensive industry experience - maybe 10 years or more. If you are thinking of going into industry as a back door into academia, realize that you are playing a very long game, with no guarantee of success.

In any case, you should expect that you will be competing against other applicants who have masters and doctoral degrees, and that, all other things equal, those candidates will likely get preference.

A Record of Successful Teaching Experience

For an instructor-level teaching position, most employers would expect that you have experience teaching at the university level. You don't have to have had experience as an instructor - it can suffice to have worked as a teaching assistant. But they definitely want to see that you have taught university-level content to university students, preferably for several semesters, and that it went well. Often, they want to see evidence of your success, in the form of letters of recommendation from faculty who supervised your teaching, student evaluations, or similar data. Enthusiasm and idealism ("I love the thought of teaching," "I have great ideas to revolutionize teaching," etc.) will not substitute for actual experience.

The most common way for people to start gaining this experience is, again, graduate school. Most graduate programs have the option for students to work as teaching assistants, often as a requirement of funding. You may not have full responsibility for your classes, but at least you are working in a university classroom and learning to address the challenges of teaching.
Your experience as a "course tutor" sounds like sort of a light version of this - a good start, but far less than a successful applicant would be expected to have.

In some cases, teaching assistants get greater autonomy, and have the ability to essentially run their own class with minimal interference from supervisors - such experience is a plus. Also, many applicants for a university instructorship will have already had experience as an instructor at other institutions; in some cases several years. So you should expect to be competing against people with such backgrounds.

I want to mention a special case of your situation - you're thinking of applying at an institution you've attended as a student. To a hiring department, this has pros and cons. Of course, they know you, and they know you are familiar with their institution's system and culture. But on the other hand, there is a benefit in bringing in people with outside experience who can broaden the "gene pool" - if they keep hiring their own people, they may get stuck in a cycle of suboptimal practices which everyone just thinks are normal. On balance, all other things being equal, I think most institutions will prefer not to hire their own students. But if you go elsewhere for a graduate degree, or otherwise get experience of the academic world outside your current institution, that would help.

On the flip side, it's easy as a student to fall in love with your undergraduate institution, think it's the best place ever, and want to work there forever. Honestly, it's probably not prudent until you have seen other places as well - your view is likely to be skewed. Also, if you did get such a job, having your entire education and career at one institution is likely to be a detriment if and when you want to seek another academic job.

Summary

If you want a university teaching job, the most straightforward first step is to earn a masters degree in your content area and work as a teaching assistant. You can then start to test the waters and see what kinds of jobs you might be able to get. But it's entirely possible that you will have a very hard time getting hired without a PhD and/or further university teaching experience.

It would also be a good idea to talk to faculty at your institution who know you well - perhaps academic advisors - and let them know this is a career that interests you. They will be more likely to have advice specifically relevant to your field and the job market in your area.