Mitchell Santner (94) missed out on a hundred for the tourists earlier in the day as New Zealand were dismissed for 310, Tim Groenewald taking five-for to go with his four from the first innings.
It left Somerset a target of 344 to win, and although James Hildreth (62no) and Tom Abell (43) game some cause for optimism, no-one else passed double figures and they find themselves just four wickets from defeat.
Resuming day three on 149-3, Santner quickly lost batting patners Ross Taylor (10) and Luke Ronchi (seven), both caught behind to Jim Allenby (3-36).
Santner ??? having started the day on 70 ??? was moving along nicely through to what would have been only the third first class century for the 23-year-old until he sadly fell six short, dismissed by Craig Overton (2-39).
At 206-6 and with a lead of only 239 at the time, the tourists were grateful to some useful lower-order contributions from BJ Watling (46), Mark Craig (20) and Ben Wheeler (33).
Craig was the first to go, edging Overton behind, before Groenewald ran through the tail to add to his double-strike from the day before for his five-for.
Johan Myburgh (four) and Tom Cooper (nine) fell early in Somerset???s chase, lbw to Wheeler and bowled by Neil Wagner.
Abell and Hildreth then added 67 for the third wicket but when the former fell to Craig (2-13), the hosts collapsed, his the first of four to fall for 19 runs.
Hildreth remains unbeaten on 62 going into the final day, but still 202 runs shy of their target, hopes of a Somerset win are slim to none.