Resuming the day at 28 for one, Reddy guided Hyderabad to 188 for four at stumps on day two.
Goa were 164 all out on the opening day after choosing to bat at the VCA Stadium.
Hyderabad lost two wickets early in the day before Reddy resurrected the innings alongside captain Subramaniam Badrinath (41) as the pair shared a 90-run stand for the four wicket.
Reddy stood like a rock at the other end and after the fall of the skipper, ended the day with Bavanaka Sandeep (18*). The opener collected 17 fours in his 177-ball innings.