Can I somehow leverage my willingness to work for a lower salary in job applications?
There is some possibility of this, but it would work only in certain restricted ways and in certain limited contexts. I don't know of any reputable college or university that, when considering applicants for a position that already exists, consider an applicant's willingness to accept less than the standard salary as a point in their favor. If you think this through, probably you can see why it makes everyone look bad.
You might have some luck getting an academic institution to make a position with an unusual combination of duties and salary that is tailored to you. Here are some possibilities:
At some institutions, faculty couples "share" positions in some fractional way: e.g. the couple together has 1.5 positions and each does 75% of the work of a full-time faculty member. If someone had come to me with this as a new idea I would have rejected it as ridiculous, but nevertheless it actually exists in the real world. I know several examples.
Similarly, you may be able to arrange a single position which is "part time" in some specified way: so maybe it is a visiting teaching position with half the normal teaching load and half the normal salary. In the very short term, such positions are common. It is not directly in the spirit of what you are asking about, but the effect could be similar.
Finally, if salary is really no object, it is much easier to get a visiting scholar position which pays no salary whatsoever. I did this myself once: the summer after getting my PhD (at Harvard), I moved back to my hometown (Philadelphia) and got an apartment in University (of Pennsylvania) City. I noticed that there was a wonderful library and gym a few blocks away from me, and I presented myself "cold" in the UPenn math department with a copy of my CV and my PhD. I walked out as a Visiting Scholar. (Nowadays I think that what I did was a little weird: why not first contact some specific faculty member in the department? Maybe I didn't know who to contact, or maybe I was curious to see if walking in cold and presenting Ivy League credentials would actually work. It did! This "position" still appears on my CV.)
If what you are interested is not covered by the above, then I would say: in general this will be hard to swing, but if you have a more specific arrangement in mind you can ask about that (e.g. here) and maybe it can be worked out.
Let me end by saying something that I suspect you already know: if you aspire to have anything like a conventional academic career in the future, advertising yourself broadly as someone who is willing to work at a discounted rate is a pretty bad idea. If you do seek to pursue this idea of yours, I would do so with much discretion.
My friend, please don't shortchange yourself. Having a modest lifestyle does not mean that you should not consider, say, buying an apartment or building a house of your own someday. And to do that you would need to save.
Also, even if you're not concerned with your own expenses and savings, remember that most of the rest of us - academics without a tenure-track position - are struggling to make ends meet with our salaries. We need to support ourselves, and often our family as well. For someone to be willing to work for less, to "undercut us" so to speak - would mean to apply a downward pressure on wages and working conditions as a whole. Now, obviously, an individual's effect on the entire system is small, but this stuff can add up; and the influence on your immediate colleagues, at the university where you will end up working, would be much greater: You would have an effect on the attitude and expectations of your department or university regarding what it offers non-tenured academics.
So, I urge you not work for cheap. If you really feel you don't need that money - consider donating it to a worthy cause (or whatever you believe is a worthy cause; it might even be donating to some scholarship fund at your alma mater).
Whilst not applicable to your case (as you're only concerned with US universities) I thought I'd add this for colour.
I've worked for 3 UK universities (currently still a visiting lecturer at one of them), and the payroll has always been done using "salary spines"; all staff of a certain level are on a "spine" - they aren't free to negotiate their own salary, so there's no pay rises unless the whole spine level goes up within the school in question, and as such there's always an explicit set salary for whatever position you apply for, with no room for manoeuvre.