Bandeep Singh scored a strokeful 58 for J&K which came off 82 deliveries and was laced with 10 hits to the fence.
Skipper Parvez Rasool made a breezy 24 off 17 balls with the help of three fours and one six before he was cleaned up by left-arm spinner Mehdi Hasan (3/55), who was the pick of Hyderabad bowlers.
A improbable task awaits J&K on the final day tomorrow as in order to avoid conceding the vital first innings lead to Hyderabad, the hosts still need to score 316 runs with just four wickets in hand.
Apart from Hasan, medium pacer M Ravi Kiran picked up two wickets giving away 32 runs.
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Earlier resuming their first innings at 341 for four, D B Ravi Teja (179) added 26 runs to his overnight score before his marathon knock was cut short by Rasool, stumped by Obaid Haroon.
Shinde struck as many as 13 boundaries during his knock before becoming Rasool's victim.
Rasool shone with the ball for J&K picking up four wickets for 99 runs.
Brief Scores:
Hyderabad 1st innings: 514 for eight decl in 118 overs (DB Ravi Teja 179, Hanuma Vihari 109, Amol Shinde 97; Parvez Rasool 4/99).
Jammu and Kashmir: 198 for six in 55 overs (Bandeep Singh 58, Hardeep Singh 32 batting, Samiullah Beigh 30 batting; Mehdi Hasan 3/55, M Ravi Kiran 2/32).