The opener scored 82 off 41 balls in a four-wicket win at Southampton.
David Willey and Sam Billings rescued England after a middle-order collapse.
The Yorkshire batsman hit a boundary-laden hundred while Moeen added 85 and three wickets.
Bairstow scored a century while Moeen made 85 off 45 deliveries in the intra-squad match at the Ageas Bowl.
Morgan explained in May that “more time” is needed before Hales is integrated back into the side.
Topley returns to international contention after a four-year absence, while Willey made way for Jofra Archer in England’s World Cup-winning squad.
Dan Lawrence has emerged as a contender.
Bairstow has returned to individual training at Headingley this month.
Bairstow and Mike Cowan were named vice-presidents.