A century partnership between Samit Patel and Brendan Taylor took Nottinghamshire over the half-way point in their 305 run chase.
The afternoon session began poorly for the visitors as they lost first innings fifty maker Riki Wessels for six – loosely playing on the back foot to edge behind.
But Patel and Taylor set about a recovery from 4-68, with both players tackling the bad balls to keep the runs ticking.
Taylor, scoring his highest total of the season, in particular needed little persuading to attack the varied Hampshire bowling – reaching 50 from 77 balls.
He eventually departed for 71, the match still yet to see anyone pass 75, when he mistimed a Mason Crane spinner to Tino Best at mid on.
Patel was not out on 46 at tea – with Notts needing 128 more runs and Hampshire needing five wickets in an intriguing final session.