With Norris taking our first spot, who's next?
Well, his fate is tied to that of "the dirty drawers" that Nauls finds. The discovery of the ripped clothing marks a point that The Thing tears through the host bodies clothes when it's assimilating them, and thanks to some logical reasoning it turns out that Palmer was the unfortunate second victim.
A few of the team are shown to be wearing long johns throughout the film, and yet Palmer is the only member who can't be accounted for out of Copper, McReady, Nauls, and Blair who are visibly seen sporting the fashionable bodysuits. Copper and MacReady don't get infected (we'll get to McReady later), Nauls gets attacked by The Thing later when responding to Garry's attack, and Blair at this point is still doing everything he can to stop the spread of The Thing.
Therefore it has to be Palmer who is attacked next, either by Dog-Thing in the silhouette scene or later by Kennel-Thing in a more aggressive and desperate manner which would explain the torn clothes.