Hampshire helped by weather
The entire first two sessions were washed out, with play only getting under way at 4pm.
An hour later, the players were off again, with the light having worsened.
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Hide AdMiddlesex took one of the seven Hampshire wickets they require for victory, with Liam Dawson gloving a lifter from Tim Murtagh to gully, where Sam Robson took a fine low catch.
Jimmy Adams weathered a testing spell from Murtagh and James Franklin to move to 47.
He was joined by debutant Joe Weatherley, who battled to reach four from 36 deliveries, before the players left the field with Hampshire 100 for four – still 236 runs behind their hosts.