We booked the Stonehenge inner circle tour. It is at 6:15am and lasts an hour. The visitor's center does not open until 9am. Is it worth sticking around to visit the center? Is there anything else to do close by that early in the morning? Any good breakfast places? We will have a car.
When we did the inner circle tour they gave us a little bag with a souvenir booklet in it, but that was many years ago. I still think that was one of the highlights of our trip. I didn't think it would be that big of a deal but walking so close to the stones and standing next to them was great. I don't know that I would hang around an extra almost 2 hours for the visitors center. Maybe if you could find somewhere to eat and then come back. I don't really remember anything in the vicinity, but maybe you could Google if no one here has an answer.
We had a morning inner circle time slot similar to yours (don't remember the exact time). Afterwards we went to the Park House Motel in Cholderton and enjoyed an excellent full English breakfast. It's on the A338, to the south of the A303, just a few miles east of Stonehenge. After breakfast we drove back toward Stonehenge and visited Woodhenge and Durrington Walls (on the A345, north of the A303). By then the visitor centre would have been open.
However, we'd already done our "general visit" to Stonehenge the preceding day. Why? Because we wanted an "orientation" and ample time to immerse ourselves in the overall Stonehenge experience. Also, we were staying in Newbury, and driving to Stonehenge in the predawn darkness -- knowing that if we were late we'd miss our inner circle visit -- would have been very stressful. We were glad we'd already rehearsed the route.
After you finish your inner circle tour you could head over to Avebury, then return to Stonehenge and visit the Center and have another walk around the circle. That would have been an option the day we did both.
The visitor centre only really makes sense before you visit the stones as it provides the history and context of the stones. seeing it after visiting the stones seems a bit pointless.
About heading over (or up) to Avebury . . . it is about 30 miles each way, and the route is on secondary roads. There are other interesting sites along the way such as the Savernake Forest, West Kennet Long Barrow, and one or more chalk horses. Avebury stone circle is open all the time, but the car park has limited hours. Then there's Avebury Manor and Garden, if you have time.
Of course if you're really a purist you could walk, LOL ...
https://www.ldwa.org.uk/ldp/members/show_path.php?path_name=Avebury+to+Stonehenge