It was only announced on Monday that Hall had rejected a new contract with Oxford and he has now agreed to link up with Town, who will compete in League One next season following their automatic promotion.
The 25-year-old is no stranger to Shrews having enjoyed a loan spell from then parent club Birmingham in 2008, scoring three times in 15 appearances.
Manager Graham Turner told the club's official website: "I am delighted that Asa will be joining us in July.
"He had an impressive spell on loan here several seasons ago and I've always been impressed with his ability and attitude to work.
"He is starting to build up a reputation for scoring long-range spectacular goals and I look forward to working with him."