You could always look to come in the back door instead of the front door by connecting with the clients first that will end up going to these clinics, instead of the clinics themselves.
When it come to athletes, someone always knows someone that is hurting & in need of therapy, if they are not receiving therapy already at some clinic. So, Endeavor to donate as much of your time as you can to some of the sports programs in your area like the local collages, University's and community events. Eventually through the grape vine you'll hear of those who are in need the most, and from working with them you'll be able to open a professional dialog with the clinics that they visit, concerning the rehab of your now mutual client.
......and from there the ski is the limit! You could either start your own Sports Massage Business & seek out referrals from the local clinics that you now have a professional relationship with or you could continue on the path of seeking employment with-in these clinics.