Spin bowler Adam Riley took seven wickets but bottom-placed Kent still had the worst of the second day at the Ageas Bowl where Hampshire finished on top.
Riley, 21, finished with figures of 7-150 having only previously taken 28 in his first-class career, but Hampshire still managed to amass a first-innings total of 455.
In reply, Kent reached 149-4 from 44 overs by the close of the second day, still 306 behind and 157 from the follow-on.
The sunshine and the perfect batting conditions provided the platform for Neil McKenzie and Sean Ervine to extend Hampshire's overnight total of 265-4 with the former racing to the 49th first-class century of his career.
McKenzie's 146 was his second-highest score for Hampshire but was ended when Riley found an edge for wicketkeeper Geraint Jones to snare.
McKenzie faced 214 balls and hit 21 fours, and it looked as if the patient Ervine would join him in scoring a century until Hampshire lost two wickets in four balls at 403.
Adam Wheater was leg-before sweeping Riley when on 21 and in the next over Ervine (86) chased a wide delivery from Darren Stevens and Jones again took the catch.
Riley bowled David Balcombe five runs later but Kent were made to wait for the end of their misery as Danny Briggs made 25 and last man David Griffiths struck Riley for two sixes before being bowled by Riley in the same over, the 146th.
When Kent responded, openers Sam Northeast and Robert Key made batting look just as easy as Hampshire had done in an opening partnership of 106 in 29 untroubled overs.
But Northeast was soon leg-before to Ervine after hitting 10 boundaries in a fluent 59 and was followed back to the pavilion by Key in the next over when the former England batsman edged Griffiths into the slips.
Griffiths broke through again at 139 when he bowled Daniel Bell-Drummond and Riley's memorable day was ruined when, as nightwatchman, he was caught at short leg by James Vince off Briggs.