Hainault joins winger Ivan Sproule, defender Evangelos Ikonomou and forward Steffen Wohlfarth in signing for the Dingwall side this month after he secured all of his work permit documentation.
But Adams has let seven players leave in recent weeks and is working on more deals before the January transfer window closes on Thursday.
Adams, whose team face Hibernian in the Scottish Premier League on Wednesday night, said: "I still think there will be a wee bit.
"We are still working in a few things, we have been for a long time and let's see what happens.
"We are looking more now in the forward area, we have probably done all our business defensively and we'll look to add to the forward area."
Hainault, 26, signs from Houston Dynamo, who he helped to two consecutive Major League Soccer Cup finals, but has been training with County for several weeks.
Manager Derek Adams told the club's official website: "Our interest in Andre has been no secret and when you look at his record that's no surprise.
"Despite still only being a young player, he's got experience of playing in Europe as well as at a top side in the MLS which is now an extremely good league which features world class players.
"Signing players from outside the EU does involve a lot more work in the background to secure the necessary clearance and visas, but this work has now secured us a player who will undoubtedly add to the squad here at Ross County."