If visiting Grand Teton National Park and staying in or around Jackson, Wyoming, in-between your hiking visits to the Tetons', you have a nice area at your disposal to do some very pleasant local day hiking just outside the Teton National Forest boundary in Jackson,WY. Numerous use trails follow in and around the perennial Cache Creek area which runs through the middle of Jackson and is several miles long. On the drive to this TH on Cache Creek Drive you will pass below one of Jackson's winter ski resort mountains- Snow King Mountain at 8005' elevation.
Please see my posted pic set/captions and posted GPS Route/notes for more details.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.