Hampshire 330 (Organ 97, Abbott 89, Bashir 3-88) and 34 for 2 trail Somerset 500 (Rew 221, Aldridge 88, Bess 54, Abbott 4-56, Dawson 4-118) by 136 runs
The visitors had slumped to 152 for eight, replying to 500, from an overnight 58 for two when Abbott strode to the crease to strike 89 not out off 152 balls, including 15 fours and a six. Organ contributed a more measured, but equally valuable 97.
The day began with a big moment for 19-year-old Somerset left-arm seamer Alfie Ogbourne, who claimed his maiden first-class wicket on debut with the fifth ball of the opening over when James Vince, on 16, edged to Tom Lammonby at third slip.
Soon it was 64 for four as the next over saw Nick Gubbins edge Jack Brooks to wicketkeeper James Rew who held a low diving catch.
At 80 for four, Somerset introduced another 19-year-old in the tall Bashir, from the River End. He wasted no time dismissing Liam Dawson, who slog-swept a catch to deep mid-wicket and James Fuller, bowled playing down the wrong line.
Hampshire were suddenly 88 for six. Ben Brown and Organ were forced to play cautiously, although Organ cleared the short boundary on the town side of ground with a slog-sweep off Bashir.
The pair put together a half-century stand in 121 balls. By the time rain took the players off on the stroke of lunch, Hampshire had reached 138 for six.
Brown fell for 39 in the first over after the interval, getting an inside edge onto his stumps to give Ogbourne a second wicket. Kasey Aldridge then had Keith Barker caught behind for 12, and the batting side were in deep trouble, still 348 behind.
But Organ had settled meticulously to his task and found a reliable partner in Abbott, who brought up the 200 by lofting Bashir back over his head for four.
Organ struck two more sixes on his way to a patient half-century, which occupied 148 balls, and the pair were still together when the second new ball was taken at 241 for eight after 80 overs.
Just before it was taken, Organ appeared to survive a sharp chance to short-leg off Dom Bess.
Abbott looked untroubled in moving to a 76-ball fifty, including 10 fours, and advanced the total to 250 with a single in the same over from Brooks.
Another Abbott four, off Ogbourne, took the ninth-wicket stand into three figures from 146 balls and by tea Hampshire had progressed to 258 for eight.
The new ball had little effect and an Organ sweep for six over mid-wicket off Bashir earned his side a second batting point before Abbott’s 14th four took the stand to 150. It was already the highest ever for Hampshire’s ninth wicket against Somerset, beating the 135 put together by Nigel Cowley and Bob Stephenson at Taunton in 1977.
Finally, with the total on 329, Somerset broke through as Organ advanced down the pitch to Bashir, who cleverly adjusted his length to beat the outside edge and provide Rew with a simple stumping.
Organ looked crestfallen as he dragged himself off the pitch, having worked so hard to keep his team in the game. He had battled away for four hours and 25 minutes, facing 206 balls and hitting nine fours and five sixes.
Abbott was also denied a deserved hundred when last man Mohammad Abbas was pinned lbw by Bess for a duck. There were only 14 overs left in the day and Somerset skipper Tom Abell enforced the following on, probably reasoning that his bowling attack could soon rest tired limbs ahead of another assault in the morning.
Abbott was sent out to open the Hampshire second innings with Fletcha Middleton and had made 15 when giving a waist-high return catch with the score on 31. Joe Weatherley edged Bess through to Rew with only a single added and Organ walked out at number four with his team back in the mire.