Gloucestershire 309 (Bracey 118, Taylor 56, Dent 50) and 28 for 1 (Bracey 10*, Overton 1-20) need another 125 runs to beat Somerset 312 and 149 (Hildreth 64, Higgins 4-29)
Gloucestershire's seam attack put them in sight of a first County Championship win at Taunton since 1993 on the third day of the match with Somerset at the Cooper Associates County Ground.
With a first-innings lead of just three runs, the home side crashed to 149 all out in their second innings in bright sunshine on a decent pitch.
Faced with a victory target of 153 and ten overs left in the day, Gloucestershire progressed to 28 for 1, losing Chris Dent, who played on to Craig Overton attempting to withdraw his bat.
At the start of the day, the visitors could add only eight runs to their overnight total of 301 for 8, David Payne offering a return catch to Marchant de Lange, who then ended Matt Taylor's career-best contribution of 56 by pinning him lbw with a full delivery.
Somerset were soon in trouble as Tom Lammonby registered a third successive duck, edging Higgins through to wicketkeeper James Bracey.
It was 18 for 2 when Tom Abell fell leg-before pushing forward to Higgins and 36 for 3 when Tom Banton guided the last ball of Taylor's first over to Dent at third slip.
George Bartlett departed first ball, caught behind off a full swinging delivery from Goodman and at lunch Somerset were in a deep hole at 47 for 4.
By the time play was paused at 2.50pm to mark the Duke of Edinburgh's funeral, Gloucestershire had struck three more blows.
Steve Davies edged a back-foot defensive shot off Payne to George Hankins at second slip and Higgins produced a good delivery that nipped back to bowl Overton between bat and pad.
When Lewis Gregory miscued off the back foot and lofted a simple catch to mid-wicket off Taylor, Somerset were in disarray at 88 for 7.
Hildreth was unbeaten on 36 when play halted and the experienced batsman shouldered major responsibility for his side's fading hopes.
At 3pm both teams, coaching staffs and umpires lined up on the outfield for a minute's silence in memory of Prince Philip.
When play resumed at 4.10pm, Hildreth and Josh Davey offered the stiffest resistance of Somerset's poor batting display.
Hildreth went to a patient half-century off 124 balls, with seven fours, and the pair had added 54 in 25 overs when he fell lbw, looking to cut a delivery from Goodman that was too close to him for the shot.
Graeme van Buuren's left-arm spin quickly sent back de Lange, leg-before for a duck and Davey was taken at first slip off Higgins for 22 to end a sorry batting effort.