How can a graduate school applicant improve his application for next year's season?
Publish.  I have seen the following pattern more than once.  (1) Applicant X applies to my department's PhD program and is rejected, not for any major flaws, but just for not standing out from the crowd.  (2) X joins a MS program at a different university.  (3) As a master's student, X publishes one paper and has two more in submission (or at least preparation).  (4) X reapplies to my department's PhD program, with a detailed technical discussion of his results in his statement of purpose and stellar letters from his new department.  (5) X is admitted in the first round, with a fellowship.

If you haven't done so already, show your complete application to the people who wrote your rec letters, and ask for their brutally honest feedback, especially on your statement.  Often students who write statements without some faculty feedback write them to the wrong audience. ("I'm smart; hire me." makes a bad statement of purpose.   "Here is the mathematics I've worked on and what I'm interested in looking at next." is much better.)

It's a long shot, but if your recommenders know faculty at the schools you applied to, you might be able to get some second-hand feedback through them.  Maybe.