Quite Possible

In November 2001 I flew from Anchorage to Dulles Airport in Virginia and then drove to Black Mountain in KY. I re-visited the KY HP the next morning in better light and then drove to the Mount Rogers Trailhead, hiked to the summit of Mount Rogers and back, drove to Sassafrass Mountain in SC, drove to the Brasstown Bald parking lot in GA, walked to the summit and back, and then drove to Cheaha Mountain, AL. I continued on throughout the south, collecting HPs and four days later I started the morning in Bellefontaine, Ohio, and visited the OH HP, drove to Spruce Knob, WV, drove to the parking area for Backbone Mountain, MD, hiked to the HP and back, and then drove to Mount Davis, HP.

So, these HPs can certainly be visited within 8 days. But your family would have to be pretty interested in visiting these HPs for them to want to accompany you on such a schedule.

I can certainly sympathize with trying to collect as many HPs as possible to maximize your HP count per airline ticket.