I used to walk to the Stringer/ Leigh Lake area and hang out and read, pretty area. Also good area to sea kayak or paddle board.
Walking around Jenny Lake was not my favorite. You can take a boat across the lake to the Cascade Canyon trailhead and have time to hike farther up the canyon. That is one that I was going to recommend. There are some great loops, but they are 20+ miles. Just hiking up the canyon is beautiful.
Death Canyon area is also cool. You can hike up as far as you like. Phelps lake is a quick side trip to jump off a rock into the lake.
Table Mountain is great, but you have to drive over to Idaho for that one. Harrison Ford did a helicopter rescue there for someone that had altitude sickness.
Dornan’s in Moose is an absolute must stop to sit on the roof and have a cocktail or glass of wine.