The apartment hunter's dilemma

In the later part of April, I will need to find a new apartment.  There's nothing wrong with the current one, but after six and a half years, successive rent increases have simply made it too expensive.  It's also bigger than I really need (two bedrooms, one of which I use, sort of, for storage).  I should be able to get something smaller but still decent and save some significant money on rent.  I'd been preparing for this for some time before my recent hand arthritis flare-up, having taken several carloads of stuff to Goodwill and also a lot to the dumpster, getting ready to fit what I have into a smaller space.

The problem is that although I've now been at my temp job for seven months, it's still a temp job.  And I suspect a good apartment complex won't be eager to rent to a prospective tenant whose employer is a temp agency.

I've raised the issue with the manager at my job (he's a good guy and I can talk frankly with him about such things), and he told me that the earliest the company might be able to think about converting me into a regular employee would be June.  And there's no guarantee of what will happen then.  It's also possible they might decide they don't need me any more and terminate the assignment entirely.

So I face a dilemma.  The obvious course would be to look for a permanent job in the next month or so.  Aside from the low likelihood of success during such a short time frame (especially at 58), I'm somewhat reluctant to leave the current job since it is, at least, a known quantity and I'd hate to give up the chance of going permanent there in favor of a leap into the unknown.  Having a good person as a manager is a major plus, as anybody who has had a bad one can attest.  And as best I can remember I've never been hired directly into a permanent job -- every job I've had started as a temp job and later led to a permanent offer.  And it usually takes several months for that to happen.

Also, it has occurred to me to wonder how true it is that nobody will rent to a temp worker.  Temp workers are not usually homeless, after all.  And I do at least have a good track record at the current place -- not one late rent payment in six and a half years.

I've done some looking on the internet for info on renting an apartment as a temp worker, but there's surprisingly little such info available -- that I could find, anyway -- and most of what I could find was specific to New York City, hardly a typical housing market.

So I'm curious.  I know a fairly large number of people read this blog.  Have any of you actually tried getting an apartment while you were employed through a temp agency?  How much of a problem did you find it to be?