Why not July... because most programs are starting the intern class that month, and it is quite busy with all the new faces and general confusion of "terns". It is likely a student would get a bit lost in the shuffle, but then again, perhaps a student may shine
The months I noted are ideal to do AUDITIONS...because if you are interested in these institutions, you will likely be interviewing with the same attendings you worked with...and they will KNOW you and your work habits (grades really mean little).
It is also a good idea to be fresh in the minds of those you interview with...(at MANY MANY institutions, the residents have ALOT of say with who enters the programs, and a recent good impression goes quite far).
LOR's are important, but the first hand experience of working hard with residents, showing genuine interest, working as a team member, and being a good guy/gal has no particular deadline...closer to the end of fall (prior to holidays) the better.