The new moon in October is probably the best time if you intend to fish during the day. The fish will be more active than usual at night during a full moon, which means they'll be a little sluggish in the day. If you're going for a full weekend of fishing and will soak bait at night either full or new moons would be good. October through mid November are prime spawn time.
Sharkathon is usually in October. If you're hoping to avoid crowds, I'd avoid that weekend. If you want to see some of the premier surf fishers in Texas chasing big sharks... Sharkathon can be a neat spectator weekend (especially for the kids).
The reds move in close to the passes in masses to spawn (make babies). This means the highest numbers of fish will be concentrated at the jetties and cuts. You'll definitely get the occasional school moving down the beach, which can equate to multiple hookups at once, but passes will have more consistent action.
Lots more on tackle, tactics, and beach tips here:
https://texags.com/forums/34/topics/2857318/1#discussionI'm not sure if you've done PINS before, but it's definitely a 4x4 required beach. I wouldn't go to far down without some prior beach experience, or lots of research.