The wing pictured is not from an ant or a termite; it looks like a moth wing and is likely inconsequential. As for the frass, we can't really tell what generated it, but would guess that ants are more likely than termites given that termite trails tend to be covered by mud tubes (and thus more hidden) while ant foraging/wandering trails are not. Carpenter ants are one possibility, though other ant species occasionally nest indoors as well. Carpenter ants favor wood that was damaged by water leaks, since it is easier for them to excavate.

If you do manage to photograph the insects, we'll take a look to try to ID them (at least between ants and termites...ID to species level is not really possible with photographs). Ants can be distinguished from termites in three primary ways: their antennae (bead-like for termites and elbowed for ants), their "waist" (noticeably pinched for ants and not discernable for termites), and their wings (all are clear, but ant wings are of uneven length while termite wings are all the same length). For both insects, only the reproductive caste has wings; all workers will be wingless. Reproductive individuals will not come to bait stations since they are not foraging for the colony, but workers can. The linked pages provide illustrations and more information, and you might need to hire a pest control professional again if either species is indeed nesting once again in the structure of the home.

Miri

Hello Jim,

Thank you for the additional images. Some definitely show an ant (doesn't look like a carpenter ant, but there are others that can nest inside a building), but we can't make out what any of the other objects might be. Termites cannot eat ants, but ants can prey on termites. The other organism doesn't look like a termite, but it's hard to tell for certain. Lots of frass and other detritus tends to be associated with Acrobat Ants, but we can't determine their ID from the images.

A pest control professional whose company has access to an entomologist (either on staff or which can send samples to a lab) may be able to ID them more concretely for you. Bait stations might be the simplest solution for eradication, though could take a little while to disperse enough of the toxin through the colony to cease activity. (Though that won't solve the potential issue of structural damage, if there is any, or just debris in a wall/floor void where a nest was situated.)

Miri

This is one popular bait brand but others might work as well. As the linked page notes, ants utilize different food sources (protein, fat, or sugar) depending on the colony needs, so sometimes it's a matter of trial and error to find a bait formulation they prefer and will visit repeatedly. We don't keep track of the various bait brands/products, nor do we make brand recommendations, so you may need to give each trial time to work and experiment until one of the various active ingredient options is successful.

Miri