Any easy solution to fishing during spawning time - be like me and fish only dry flies. I doubt if any fish that's rising is also reproducing.
As for when things really start to pick up, for rising fish, usually when the weather starts cooling down noticebly. From my experience, this starts generally around mid sept, and stays good thru oct. Again, I'm talking about fall hatches, and the fish starting to rise thru out the day again like they do in early spring, instead of just early morning and evening. I refer to this as their fall mode - mostly fishing to autumn olives, but lots of streams also have good slate drake and caddis hatches. And tricos will still be around thru oct on some streams also.
Fall has become probably my favorite time to fish. There are no crowds to contend with - most outdoorpersons thoughts are turning to hunting by then. You will likely have most streams to yourself
Those annoying mosquitos - which have particularly bad this year where I've been fishing - start to disappear.
And again, those beautiful fall hatches. On streams with tricos, I frequently find fish rising all day. The tricos usually go all morning then, and just as they peter out in early afternoon, the olives start, and last pretty much the rest of the day. Doesn't get much better than that!
I recommend you get out and enjoy it.