Gloucestershire have lost on their last eight championship visits to New Road and when Jones claimed Grant Hodnett for one and Craig Spearman for seven, it looked like they were on for more of the same.
Gareth Andrew followed this up with the wicket of Hamish Marshall, and Steve Magoffin did more damage as he knocked out Marcus North's off-stump and had Alex Gidman caught behind.
But this brought the centurions together, and they stemmed the flow and took control of the match.
Snell added 124 through the afternoon, while Taylor began to punish Gareth Batty hitting him for three sixes before he was caught out.
This did little to dent the Gloucestershire charge, however, and Hardinges and Snell put on 104 before stumps.