Maybe you should figure out where you're going first, and then the timing of your move there. If you're moving to a metropolitan area it will probably be easier to find temp work, but in my experience you can't just show up and get a job the same day (unless you're a day laborer!). My friend had to do several interviews before she was given any decent positions. Also, even if it may not be easy financially, it may not be a bad idea to stay unemployed before law school. It will only be for like 6-7 weeks, since schools generally start orientation the 3rd week of August. If you start saving now, there's no reason not to have 2 months of living expenses saved up before the end of June.
Also, if will also make for an easier transition if you have some time to familiarize yourself with the area. For example, I would like to move to Ann Arbor and get settled 2-3 weeks before classes start and not work, but get acclimated to the area, but I have to move 3 days before orientation starts. If I could, I'd want to spend some time finding out grocery stores, restaurants, bookstores, etc. before launching into the stress of 1L.