In the week when we first had the cat, before he got out, he was definitely starting to get marginally more comfortable around the flat. Well at least to the extent that he would explore at night, and one evening he came out of the conservatory and into the lounge, where he hid underneath an armchair, but then came out and walked around the edges of the room, staring at me the whole time. This seemed like a big step forward! And then he got out, and now we're back to square one.
Yes, it definitely makes sense that we should give the cat to be rehomed now, or keep him forever, not let him get settled in then make him leave!!
Mate.
You're going to see fuck all in a week.
Absolutely fuck all, beyond the very earliest glimmerings. At best.
I'm not sure how old your cat is, but it's been through three homes and a bout of homelessness AND a cat shelter in the last... what? 6 months?
The poor thing needs a bit of stability before it can even begin to show the 'personality' beneath the scaredness and confusion. And, not being funny, but who the fuck are you to the cat at the moment? Someone who it was vaguely around for a week, and who then spent a fair while scaring the shit out of it when all it wanted was food, and who's now locked the poor bugger in.
And
still it came back for food and warmth. Against its evident fear and confusion.
Cats can be truly awesome. Like, we've had our Nigel for about 18 months now (rehomed after her owners were arrested for domestic assault, and sectioned following domestic assault respectively; with the woman then bringing a guard dog into her territory). And it's amazing how she's adapted to us, and we've become a part of her world. So to speak. Step by step and increment by increment, but suddenly noticing how much more *adapted* to us, as her people, she now is.
It sounds as if your cat may never be a lapcat, or without a degree of fear / anxiety; but... if you're going to judge that on the basis of a week... that's... massively on the wrong timescale. And massively unfair on the cat.
tbh, it does sound as if you might need to adjust your paramaters for assessing cats, or rehome the poor bastard pretty swiftly. It's had a monumentally fucked up time already, going by the sounds of it. If there's going to be even more turmoil in the near future, then ffs get it done.