I believe you can hunt the river, but it has to be north of where spring creek and the San Jac meet (Montgomery County). Definitely narrow, and all private property on both side owned by development and Sand plants.
Yes, you can hunt most navigable waterways, but there are state laws that can make that difficult contained to a narrow waterway like the San Jacinto. The one that your projectile can't cross a property line already mentioned. As well you can no longer hunt flooded water over private land. Then there is the problem with ducks or cripples falling onto private land off the river. Then you have to choose which regulation to break, either trespass or waste of game. But if you find an area wide enough and or adjacent landowner blessing and or game warden blessing, go for it.