Thompson was seen on Wheatsheaf Road, Wolverhampton, at 01:00 GMT on Monday, before then being sighted on the school grounds at 07:15.
He was last spotted wearing a distinctive camouflage jacket, grey gilet, jeans, and a beanie hat. He was also carrying a red rucksack and may have been wearing sunglasses.
Supt Martin Hurcomb said: “Our teams are continuing to investigate the circumstances around this tragic incident and we are offering support to Mr Thompson’s family.
“Inquiries are moving ahead as investigators work to piece together his movements and we’d ask again that anyone who may have seen Mr Thompson in the early hours of yesterday does get in touch.
“Neighbourhood officers are also out in the area, so please do speak with them as they can offer reassurance to anyone who has concerns.”