I'd start looking as soon as you know which school you're going to attend, you don't have to sign, yet knowing where you're going to be will help with the stress of starting school in August. The Health Professions complex filled up last year before second look day, and graduate housing can be quite difficult to get into. You have a chance at the Health Professions as many of the M2's living there now will be moving out in early summer. A lot of the other complexes will still have availability through the summer. I signed in May with my building, and was given my apartment assignment in July (had to wait for one to open up).
As to the avoiding those 2 months of rent in the summer... you could potentially sign a month by month lease, but that will likely end up costing you more in the long run. You will have optional research opportunities with the school during that summer, so you may want to keep the apartment. Honestly, most students just have to swallow that expense, yet there may be a way around it if someone else has a way.
Hope this helps!
p.s. anyone else having issues with this site? It looks all crazy right now!