Hampshire 79 and 110 for 4 (Overton 4-27) trail Somerset 336 (Overton 74, Abell 64) by 147 runs
Rain and bad light frustrated Somerset's push for victory with just nine-and-a-half overs bowled before play was abandoned after just 45 minutes late on Saturday at the Ageas Bowl.
Hampshire closed the day on 110 for 4 in their second innings - still trailing Somerset by 147 runs - with opener Joe Weatherley unbeaten on 34 and skipper James Vince 5 not out.
A heavy overnight downpour in Southampton left the outfield saturated as ground staff worked from 10am to dry off the standing water before play finally got underway at 5.30pm with a minimum of 16 overs to be bowled.
Weatherley and Sam Northeast came out to bat in perfect seam bowling conditions, their side on 92 for 3 as heavy clouds enveloped the ground and with the floodlights on.
But it was the excellent Overton who finally made the breakthrough with his fourth wicket of the innings, as Northeast mistimed a push towards mid-wicket and George Bartlett took a stunning one-handed catch at short-leg.
However, the visitors' joy was short-lived as play was abandoned by umpires Graham Lloyd and Alex Wharf due to bad light.
The forecast is better for Sunday with Somerset looking well placed to consolidate their second-place spot in Group Two after leaders Gloucestershire beat Middlesex at Lord's.