The town of St. Regis Falls stocks the river of the same name (I think it's the east branch, but whichever branch runs right through the village) at the park. There's some access off some of the back roads (one's a dirt road to the east about a 1/4 - 1/2 mile north of the village). St. Regis Falls is a little over an hour away.

There's also some trout on the upper reaches of the Grasse River, but I've never fished there.

If your feeling really rambunctious, you could drive to Lake Placid and fish the W. Branch of the Ausable.