Some ideas -
"Small" actions by the protagonist that, presuming they're oblivious to the actual threat (or it's so abstract as to be unable to be determined specifically by the protagonist or perhaps even the reader), make "logical" sense but are essentially the "wrong way to go." Ego-driven main characters who operate as if they are the only ones in the world are easy to write like this, although coming up with the small, logical actions may prove difficult.
Also along those lines, the character might "think" they've determined what the threat is and act accordingly, but be absolutely wrong or have misread or underestimated it.
Of course, the alternative to the first option is the classic "suddenly becoming aware that they're in deep shit and freaking out", with the proviso here being that "freaking out" may just mean automatically resorting to flight (which may be impossible) or fight (which may also be impossible) - but ping-ponging between these two can move a narrative forward - or just breaking down in the face of the threat and waiting for it to reveal itself.
James characters tend to have some intellectual goal that drags them forward, Campbell's tend to be driven by their character/psychological faults and Aickman's...well, it's the same as Campbell's but usually buried under a thick skin of manners and cultural clutter. In fact, that's another exploitable element, one at which Aickman was a master - the accrual of small events signifying that something is wrong but contrasted against a raft of cultural and societal niceties that are slowly unwrapped from the character. Stories like this *tend* to be British (simply because the Brits really developed an enormous cultural carpet bag of the stuff) but there's no reason it couldn't work with other cultures, with enough planning (which is one of the reasons why the "fish out of water" set-up is so popular)