How to maximise one's chances of getting a good postdoc position?

I am also in your current situation and look for the same. Here is something I have learnt so far which are important to get postdoc job in good quality institutes. Here is the list without order:

• Publish in under-spot conferences and seminars
• Reference letter from pioneers of the domain (if you can liaise)
• International collaboration, research, and publication
• Face-to-face visit. If possible try to make an appointment with potential postdoc supervisors and visit them in person. It helps both of you to better evaluate and decide.
• Search among friends of your friends for an open position. I use Microsoft Academic portal "http://academic.research.microsoft.com/" to search co-authors of my supervisor or academician friends of mine to see if there is any potential postdoc supervisor in their network. So you can give a try and if found, ask your supervisor or your academician friends to play a role and introduce you to potential boss. I think it should work well.
• Upload your papers everywhere you can (take care of copyright issues) and try to increase their visibility and citation; the higher citation, the better chance.
• Make your professional account in Google Scholar and keep that up to date.
• Enroll in academic organizations like IEEE and get membership. It gives lots of benefits and is like a mark of attachment and care to the society.
• Collaboration invitation. Try to prepare a paper and get in touch with potential postdoc supervisors to invite them collaborate. This can be a venue to exchange couple of emails and get to know each other more. You can later use this opportunity to request for position.
• Try to get chance to visit potential labs as visiting scholar. some institutes welcome visiting PhD students under different schemas like student exchange or international collaboration.- Volunteer job in varied community; not necessarily academic environment. Try to show your passion to work independently in every workplace regardless of the details.
• Online Connectivity. Active participation in online networks like Linkedin, Researchgate, stackexchange(here), and so on.
• Patent is also important since institutes are increasing concern about patents and desire their postdocs to produce patent beside publication. It also enhance your industrial career too.
• Keep sending application for advertised positions and don't get disappointed.
• Keep sending email and follow up to potential postdoc supervisors with potential postdoc projects (it depends on your field) even if they have no current position. Your messages may impress them and motivate them to hire you to work on a project you define. If yes they can apply for funds (I know one of my friends in CS got offer in this way).

I hope these points help you better hunt a good postdoc in good research group.

EDIT: Patent is modified to address some concerns on wording. EDIT2: Some change in the order to better highlight their importance.

My feeling, in the limited context of mathematics in France, is that the research record is by far the most important thing. Of course, one should do its teaching duty carefully (edit because it is one's duty, but also because being known as a very sloppy teacher can close some doors), but all other aspects are tertiary.

[Paragraph edited upon further reflexion] I would only mitigate this in favor of the visibility issue: good research needs to be shown in order to give you benefit, so you should have an up-to-date web page, post your preprints on the arXiv, give talks whenever given the opportunity, and in some circumstances propose one. Also some networking may be needed, in particular if you advisor does not network for you. This means for example seizing the opportunities to collaborate with more advanced researchers, send by e-mail your best work to a few people who you think might be interested, etc. Don't overdo it though.

I would definitely advise against spending time on administrative things as grant funding and the like. Participate in a collective grant if offered, but that's not a PhD job to do the paperwork.

In addition to the points already made, something that hasn't been explicitly mentioned yet is to let your colleagues know that you are looking for a job. Everyone in your immediate lab group and advisory committee should know your name, what your specific research interests are, and that you're looking for a job. Most immediate department members should be aware of your general research area and that you're looking for a job. When you network, you should specifically mention that you are looking for a job and to "keep an eye out for any positions". Your personal web space should state that you are looking for an academic job and to contact you.

This might be rather obvious in the context of the question but I thought it was worth repeating since this is a very concrete and specific thing you can do to improve your visibility. Many people say "network" or "visit departments" or "publish papers" and these are essential to getting a job in the long term, but if you don't mention to all of these people that you're looking then they're not going to keep you in mind when they do see a position.

Summary: if you network, tell people "I am looking for a job" and then ask them to "send along opportunities that may be interesting".