Overton - the leading wicket-taker in the LV=County Championship - demonstrated his batting credentials by top-scoring with 74, then starred with the ball as his side opened up a 165-run lead as Hampshire closed on 92-3, writes IAIN McCULLOUGH.
The England paceman hit 10 fours on his way to his 13th first-class 50, before he removed opener Ian Holland, Tom Alsop and nightwatchman Kyle Abbott during a hostile spell of bowling late in the final session.
Alsop and Weatherley had dug in admirably in a 56-run stand before the left-hander fell for 32 five overs from the close when he lobbed an easy catch to Josh Davey.
Overton struck again in the penultimate over of the evening when he trapped Abbot lbw for a duck for his 30th wicket of the campaign.
Weatherley was unbeaten on 31 alongside new batsman Sam Northeast, with the pair to resume their innings at some stage on Saturday with heavy rain forecast.
It seems only the weather can prevent Hampshire from sliding to a second successive Championship defeat.
Somerset started the day on 142-5 and added another 47 to their total before skipper Tom Abell's stubborn innings was ended on 62 by a ball that nipped back into the top of his off-stump from Abbott.
Nightwatchman Jack Leach, dropped when on three in the final over of the first day, made the hosts pay for that mistake with an entertaining 34.
But the England spinner perished after trying to lift Keith Barker over the slips cordon only to find the hands of Liam Dawson.
Somerset's deep batting line-up then proved its worth as 143 runs were put on for the last three wickets, spearheaded by Overton.
He was ably supported by Steve Davies and Lewis Gregory who each scored 33 as Somerset, who sit in second spot in Group 2, one place above Hampshire, built up a handsome 257-run lead after recovering from 36-3 to be 336 all out.
Left-arm veteran Barker was the pick of the home bowlers, finishing with 4-67 on his first appearance since last August, while Abbott took three wickets.
Barker said: ‘There is a lot to do. We have really tried to apply ourselves in this innings. It might not be doing as much as on day one, but there is still a bit there in the wicket, we just have to dig deep now and see where we end up tomorrow.
‘Joe (Weatherley) has grafted hard and we all need to do that and don't make it easy for them by giving our wickets away.’
Overton, who last played Test cricket in September 2019, remarked: ‘I feel I am still not quite at the top of my game - there are still parts I am working on.
‘I think as a cricketer you always want to improve yourself and that is what I am doing. But overall I am pretty happy with where I am.
‘Being in the Test team is the aim, every cricketer wants that. I have played in four Tests and hopefully if I keep doing what I am doing you never know what can happen.’