So, you're basically out of luck, as the pros do not compete outside the tournaments. You'd need to catch an exhibition or temple game, but I'm not aware of any handy schedule for these. Edit: Found a list on Japanese Wikipedia, but the regular ones listed there are only in February, June and October.
What you can do, though, is head to Ryogoku in Tokyo and check out training at a sumo stable. The easiest option is to join a guided tour (any travel agency can arrange this, Google around), or you can roll your own and contact eg. Tagonoura Stable directly (English spoken!). Be aware that you're expected to "donate" around 1500 yen/head even for a nominally free tour, since you are (to some extent) getting in the way of their work.