Hampshire 57 (Quinn 6-23) and 105 for 4 need another 273 runs to beat Kent 165 (Compton 63, Abbott 4-46) and 269 (Leaning 112, Abbas 4-68)
Jack Leaning celebrated his second century of the season to boost Kent's hopes of avoiding relegation from LV= Insurance County Championship Division One as they set title-chasing Hampshire 378 runs to win.
Stand-in captain Leaning struck 112 as the pitch became more conducive to batting, following 23 wickets falling on a fast-forward opening day, as Kent totted up 269 runs.
It meant Hampshire, who had seen Mohammad Abbas claim four for 68, required the fifth-highest chase in their history to keep themselves in the title race - with Surrey highly likely to pick up a win and six bonus points.
Joe Weatherley was exceptional for his unbeaten 54 but Hampshire lost openers Felix Organ and Ian Holland plus nightwatcher Keith Barker and Nick Gubbins while knocking 105 off the required total in 35 overs - with 273 runs still required to victory.
Having taken three evening wickets - albeit two were nightwatchers and the light had dimmed - Hampshire were optimistic they could blast the remaining seven Kent batters, have a modest target to chase and reignite their title tilt.
They were in for another chastening day, as the pitch somewhat flattened out and their visitors took advantage.
Tawanda Muyeye and Daniel Bell-Drummond came out with a game-plan to attack with the pair driving with glamorous abandon, the pair putting on a carefree 35. Muyeye gained a life on 34 when dropped at third slip but failed to use it when leg before to Kyle Abbott in the following over.
Bell-Drummond continued on his way to 40, with all but six of the runs coming in boundaries, with 51 added with Leaning before he was bowled playing around James Fuller. Ollie Robinson also fell before lunch edging a slog to first slip.
The afternoon saw Leaning come into his own. He set up outside of his crease and then lurched further forward to try and get to the ball before it significantly moved.
His hypothesis worked as he comfortably collected runs, most often guiding to third. His fifty came in that region, although through an edge between wicketkeeper and slip, in 89 balls.
Leaning is on course to average above 40 in his first two full seasons at Kent, since moving from Yorkshire, but last season his failure to convert blackened his 745 a tad. He turned a fifty to a century just once in seven attempts in 2021.
He has almost matched his overall tally from last year, up to 714 after this innings, and has now scored two centuries - this being in much trickier conditions than his 128 against Gloucestershire at home.
The right-hander reached his eighth first-class century by advancing and dispatching a six over midwicket. He had batted through over an hour after lunch with Harry Finch for 78 before the latter was lbw to Abbott.
From there wickets fell more regularly as Leaning scored all but one run off the bat in stands of 17 and 21 with Joey Evison and Nathan Gilchrist. Evison was caught behind, Leaning upper-cutting to deep point and Gilchrist picking out long on to end the innings.
Only Hampshire sides in 1983, 1985, 1990 and 2006 had scored greater than 378 runs to win a match. Chances of replicating those performances got off to a hitch when Holland was leg before to Matt Quinn in the fourth over.
Organ and Weatherley bravely copped knocks on the hand during a ferocious Conor McKerr over with the homegrown duo putting on 68.
But after back-to-back boundaries, Organ edged Harry Podmore behind. He left the middle banging his bat against his helmet in frustration before Barker starved off 17 deliveries and then looped a bouncer to midwicket. Quinn picked up his eighth wicket of the match as Gubbins left a delivery that nipped back into his off stump.
The smattering of wickets meant Weatherley shyly celebrated his 94-ball half-century - his first since April.
