With 8-10 hours, might try hiking up the Glacier Trail to Garnett Canyon and on up to the saddle between the Grand and the Middle Tetons. Or as far as you can get before time remaining (or snow level?) causes you to turn around.
When last up there several decades ago, the trailhead was at Lupine Meadows parking lot. We started late morning, arrived at the saddle late afternoon, and spent the night there before returning to parking lot.
Partway up the switchbacks along Glacier Trail to Garnet Canyon (little more than four miles from trailhead), can forget Garnett and the saddle and instead take a fork to the right that leads to Surprise Lake and on upward to Amphitheater Lake high above and north of Garnett Canyon.
Maybe someone here has more recent information on the above locations and current conditions. Could also call the ranger at Jenny Lake for trail information and to see if a permit is needed, even though you are only planning to go on a day hike.
The following description of the route is from GORP:
"8. Lupine Meadows.
"Amphitheater Lake 9.6 mile round-trip, 8 hours, 2958 ft. elevation change. Strenuous. Climb up to glacial lakes surrounded by subalpine meadows. Horses not allowed.
"Garnet Canyon 8.2 mile round-trip, 7 hours, 2160 ft. elevation change. Strenuous. Trail leads to the mouth of Garnet Canyon. Horses not allowed."
See also these links for more info: