What pringle89 is describing is a process of desensitisation which is shown to work, provided (and this is the critical bit) that the dog is never left for longer than it takes to become anxious.
If the dog can tolerate 30 seconds alone without becoming worried, then that's the point you start. If it can tolerate 1 min you start there. If it can only tolerate 10 seconds you start there. If it can only tolerate you looking at your keys, then that's where you start.
Once that starting stage becomes a total non event then you can raise the time by a few seconds. Creep it up in slow enough increments that the dog has time to desensitise (get totally used) to each stage. If the dog ever shows signs of anxiety then you must go back down to a stage they are fine with.
If you 'practise' at stages the dog is anxious about you risk sensitisation - which is the exact opposite of what you want.
The stage a particular dog can cope with will be personal to that dog. However, it will be affected by other stresses in the dog's life. A rehomed rescue dog is going through a really stresful time so it may be that they are especially sensitive to being left at all while they settle in.
If it were me/mine, I would try and find a way the dog was not left at all for a few weeks while he settles in and settles down. Then start a process of slowly building up the time as above. A camera can help you see what happens when you are gone and it's worth knowing the signs of stress in a dog because by the time he is barking, the chnces are he is very sressed indeed. As a result barking or not barking is a blunt measure to determine whether or not the dog is coping.
To leave the dog on Thu is to gamble:
a) He might be fine
b) It might escalate his anxiety and behaviour
I suspect b is more likely than a, so I wouldn't be in a gambling mood myself.