Richard Gasquet was pushed before reaching the Winston-Salem Open third round, while Jiri Vesely won through his opener.
Frenchman Gasquet, the ATP 250 tournament's top seed, battled past Brit Daniel Evans 6-4 4-6 6-3 on Monday.
World number 15 Gasquet managed to take his chances in the deciding set, converting one of his two break points as Evans squandered both of his.
Awaiting Gasquet in the third round is Stephane Robert after the Frenchman beat Federico Delbonis 6-4 5-7 6-4.
The only other scheduled second-round match did not go ahead, as Mikhail Youzhny handed fellow Russian and 12th seed Andrey Kuznetsov a walkover.
Youzhny had earlier been challenged by qualifier James Duckworth, claiming a 7-6 (10-8) 1-6 6-3 victory.
Vesely came through his first-round encounter against teenage Dutch lucky loser Tim van Rijthoven 3-6 6-3 6-3.
Australian John Millman battled past Yoshihito Nishioka 4-6 6-3 6-4 and Brit Aljaz Bedene was also tested, beating James McGee 3-6 7-6 (7-5) 6-2.
Others had fewer problems as the likes of Malek Jaziri, Guillermo Garcia-Lopez and Taylor Fritz progressed.
Jaziri brushed past Radu Albot 6-2 6-4, Garcia-Lopez beat Damir Dzumhur 7-6 (7-0) 6-1 and Fritz won a battle of teenage guns by overcoming fellow American Frances Tiafoe 6-1 6-4.
Rajeev Ram sent down 13 aces but suffered a 7-5 7-6 (7-5) loss to Kyle Edmund, Lukas Rosol won a rollercoaster clash against Victor Estrella Burgos 6-4 1-6 6-1, Diego Schwartzman thrashed Mikhail Kukushkin 6-0 6-2 and Donald Young had few issues beating Pierre-Hugues Herbert 6-4 6-1.