Just wade out to the first sandbar and cast a silver spoon at dawn off the beach for specks, occ redfish, lots of ladyfish, and an occ blue fish or blue runner and a rare jack.
Fun fishing and catching, but don't be dead set on a fish dinner.
I have never fished the Little Lagoon, except on the cut in GS where it empties into the gulf. Lots of water movement in a very small area you can wade and fish.
There is a parking area. Head west into GS until it almost dead ends, you will cross a small bridge over the cut. Or you can fish an outgoing tide there for redfish in the gulf as water leaves Little Lagoon.
Do you have a kayak? Not sure if you can rent them, but I have heard you can really get into the specs on the west end of LL early in the morning with topwater baits from a kayak.