What can I do at home to complement my sessions?

My gut reaction to this question is CARDIO.

That is the answer for people that are coming to pilates at least twice a week. If you do not get to the studio that often or have blocked off enough time to do more than you daily cardio. I would take a look at a few of the ebooks and videos that are on the studio site.
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Exercises You Can Do on the Road!

Another option is took for some good videos to supplement your workouts. I like to send people to videos and podcast from teacher training programs. These are as close to authentic pilates you are going to find on a tape. I like the podcasts from Balance Body and our youtube channel.