The possibilities are almost endless, and Richard and Mike have given you several really good choices.
Here is one more. Go to Pinedale, Wyoming and drive west to Big Sandy Campground ( one of many great trailheads) and hike in to Big Sandy Lake, go above to Clear Lake/Deep Lake, come back down to Big Sandy Lake and cross Jack Ass Pass to Lonesome Lake/Cirque of the Towers.
You can do numerous loops in this area to take up a couple of weeks.
Or, you can just catch the Continental Divide Trail again out of Big Sandy Lake and hike north west 50 to 100 miles or so along the west side of the Continental Divide, over 2 weeks to a month. If you are a trout fisherman, there are literally many hundreds of trout filled lakes, plus streams and rivers.
Only thing is you should be a very experienced hiker if solo, or be with some one experienced. You don't have to rock climb if you stay on main trails, but these are jagged high altitude mountains in deep wilderness. Depending on time of year, it is not uncommon to hike all day and not see any one else, and to have one of these trout filled lakes all to yourself. And what time of year? Regardless, you must be prepared for freezing weather and snow every month of the year. I have seen 24F and 1+ foot of new snow in late June, though weather is usually not too bad in the summer. Usually lows of 40, but could be a lot colder. Elevations will be from 9000 to 11000 ft if you don't climb the nearby 12000 to 13000 ft peaks.
But if you are up for it, I think you will never forget this place. Take bear spray just in case.