Derbyshire 257 (Palladino 60*, Briggs 5-50) and 31 for 0 require another 315 runs to beat Hampshire 388 (Wheater 87, Smith 82, Bates 50, Palladino 4-97) and 214 for 5 dec.
Captain Jimmy Adams hit his second half-century of the match and then made an attacking declaration to leave the final day at the Ageas Bowl nicely poised.
Adams, having scored 60 on day one, made 67 to help Division Two leaders Hampshire reach 214 for 5 at which point he declared with 14 overs left in the day, setting Derbyshire 346 runs to win.
Derbyshire survived until the close with Stephen Moore on 22, including three boundaries and a six, with Paul Borrington 9 not out.
Derbyshire resumed the third day at 166 for 7, trailing by 206, and were soon in more trouble when Kyle Abbott removed David Wainwright for 1 but the visitors at least managed to avoid the follow-on before Tim Groenewald was run-out at 245 for 9.
Left-arm spinner Danny Briggs claimed his five-wicket haul when Mark Footitt was stumped and Derbyshire were all out for 257.
Hampshire reached 13 without loss at lunch but shortly after the break Will Smith was caught behind off Tony Palladino for nine. But skipper Adams, though, soon moved the hosts' advantage over 200 before Joe Gatting's promising knock was ended on 39 when he was caught in the slips as Hampshire reached tea at 106 for 2.
After tea, Adams stepped up the run-rate, quickly bringing up his 50 up in 117 balls, and supported by James Vince, the hosts went on to 153 before the Hampshire captain was caught by Borrington off Wainwright. His 127-ball innings included six fours and two maximums.
Vince's 40 and a quickfire 32 not out from Sean Ervine - with four boundaries and two sixes from just 13 balls - helped the home side declare before Adams put Derbyshire back in for the end of the session. But there was no late breakthrough as Moore and Borrington batted out the closing 45 minutes.
Briggs was the pick of the Hampshire bowlers in Derbyshire's first innings, returning 5 for 50 off his 28.3 overs, which included 10 maidens, to give his side every chance of a fourth Division Two win if the weather holds on Wednesday.