Jonathan Walters gave the visitors a deserved first-half lead, before Joey O'Brien secured a share of the spoils for the Hammers after the break.
The draw moved Tony Pulis' side onto 13 points from 12 league games, just three points behind Swansea City with back-to-back home games against Fulham and Newcastle United respectively to come.
And the industrious Scottish midfielder thinks they can still improve.
“I thought we were absolutely fantastic during the first half of the game at West Ham,” he told the club's official website.
“It was arguably the best we have performed all season.
“We maybe should have scored another goal or two and put ourselves into a position where we almost certainly would have gone on to win the game, but we weren’t able to do that and West Ham came back into it.
“When you have players like Kevin Nolan and Andy Carroll in the side then they are always going to be a danger and they pushed us back during the second half. I think we deserved the point we got though. We had the first half, they had the second.”
Tony Pulis' side have endured a difficult start to the season, having faced all of last season's top four in addition to Brendan Rodgers' Liverpool.
|POTTERING INTO THE TOP FOUR?|
|200/1||Stoke are 200/1 with William Hill to finish in the top four this season|
They have, however, lost on just three occasions so far and held both Arsenal and the Reds to a goalless draw, either side of a 1-1 draw against Manchester City.
Both Chelsea and Manchester United inflicted defeat on the Potters, in addition to Chris Hughton's resurgent Norwich, but following an impressive display at Upton Park, Adam is hoping Stoke can string a run of results together.
“We need to take the performance from the first half into this weekend’s game with us," he added.
"We aren’t too far off picking up a string of good results, and I am certain that if we keep doing what we have been then we will be where we want to be come the end of the season.”