Poulter had further reason to feel aggrieved after a mix-up saw him informed that he had done enough to qualify ahead of his quarter-final at the WGC-Dell Technologies Match Play, only to be told 10 minutes before his defeat to Kevin Kisner that he required another victory.
Poulter had begun on Thursday with a one-over par 73 but then stormed through the field on Friday in signing for an eight-under par 64 that moved him to within four shots of the second day 11-under par lead.
The Houston Open offers one final opportunity for PGA Tour players to secure a last-minute ticket to the upcoming Masters - the victor advancing to Augusta.
Chalmers had eight birdies - five in the first eight holes - before a bogey at the 17th.
The 29-year old Fowler headed to the scorer's hut trailing a stroke behind compatriot Ben Hossler (68) and tied in second place with fellow countrymen Abraham Ancer (66), Sam Ryder (68) and Nicholas Lindheim (66).
At the ANA Inspiration, the first major of the season, Jodi Ewart Shadoff shot a 67 to lie five off the clubhouse lead, set by South Korean Park Sung-Hyun.
Spieth said he had found something with his swing after struggling recently.
"I'm hitting a lot of the shots that I will use on a few holes next week, these low running slices", said the 2011 Houston champion. But I don't have to (shoot) 8 or 9 under.
"It's been a while to post two numbers like that and obviously I'm going to need to go and do another one tomorrow".
Hossler and Spieth are frequent practice partners since Hossler has settled in Dallas. Still, Spieth is familiar with Hossler's game.
Phil Mickelson was at 8 under heading into the 18th until a triple-bogey derailed his round. Both played for the University of Texas, but not together. It's no surprise that he's (on) top. The victor Sunday will join them, or even formerly exempt.
"I actually told my caddie on the range yesterday (when I was) hitting the ball all over the place and not making solid contact, I was like, it's amusing", Hossler said.
"I made a brief adjustment Saturday afternoon at Bay Hill and putted as confidently as I have in a long time, then brought it in this week", Glover said.