I know some guys that will soak their birds with several water changes in an attempt to draw out / reduce the injected solution before putting the bird into their own brining solution. They claim in makes a difference, but to me, it sounds like too much trouble. I just go ahead in put my solution injected bird unto my brine mixture, let it brine for 24 hours, rinse it off and put it into the smoker. I've never had an 'over salty tasting' bird yet. The way I understand it, when the sodium content of the bird matches the sodium content of the brine, the bird quits taking on any additional salt.
YMMV~
Enjoy the Smoke!