I am so glad he survived the wet tail with prompt care from you and the vet
He is a little fighter. As someone else mentioned - now is still a time to avoid too much stress. His new cage is amazing and I also love the pink wheel
However a cage move is also a very stressful experience and the same advice applies with a cage move as when they first come home, which is to leave them to settle in for a few days and to not clean anything out or change things around etc for the first two weeks. During that time they are adjusting, scent marking everything, and trying to feel at home.
They can often be very excited the first day and then have a funk when they realise they're not going back to the old cage. It's usually advised to move all the old bedding across to help it feel familiar, so if you haven't and still have some of it then adding it on top of the existing substrate could be a good idea. If not, don't worry too much if he suddenly gets a bit nervous and shaky during the next week. After about 10 days he will be happy as larry in there. Some adjust quicker than others.
They scent mark everything partly to "claim" it and partly to find their way round the cage and know where everything is (they don't see well at all). Hence it being important not to clean him out for the first couple of weeks as it removes all the scent.
If it gets particularly whiffy after a week you could just "spot clean" the pee area - often a corner of the cage which they choose as a toilet to pee in - ie just take a handful or two of wet substrate out (doesn't have to be too thorough) and put a new handful or two in and mix it in with the old.
Best tip I can give is to use a litter tray - they will use it if you put it in their chosen pee place and then you need to do clean outs much less often - you just empty the litter tray every 5 days or so. Best thing to put in the litter tray is Chinchilla bathing sand (most people get the Tiny Friends one which is easily available). It soaks up the pee.
Also moving things around when they've just scent marked them can stress them. So at this stage best to leave everything as it is and avoid the temptation to "tweak" things!
However adding extra substrate should be fine as they like that, adding a litter tray is fine too, or adding a new toy for example is fine - as long as it doesn't mean removing or moving something else.
Cleanouts do stress them generally, although they get more used to it as they get older and have developed a bond and trust with you.
With a cage like that and the base filled with substrate you can easily go 6 to 8 weeks or even longer without needing to do a substrate clean, especially if using a litter tray. Even then I tend to only replace about half the substrate, replacing the bottom clean half and mixing it in with the new so it still smells a bit familiar.
Also best to only do "partial" cleans. So anything else that needs cleaning should be done at a separate time to the substrate. eg the wheel a different week, and any toys a different week again. Toys tend not to need cleaning that often. And also best to always leave the nest and hoard, unless pee'd on. If they do get pee'd on (occasionally young hamsters have accidents!) then try and leave a bit of dry nest and a bit of dry hoard behind and always put new food out to replace the removed hoard in exactly the same place. And a big pile of new nesting material in a heap somewhere in the cage (not actually in the nest or house - they like to forage for it and arrange it themselves).
Removing the nest and hoard can lead to some abnormal behaviours and they will start peeing on it regularly (to deter people from stealing it!). There's all sorts of advice to get round that if it happens to get things back to normal!
They are happy when they can have normal behaviours which is to nest in a dark place, bury hoards, dig and burrow down in substrate, forage a bit and have an appropriate cage set up with enough space and enough variety.
So basically - the right size cage (which you have) a shelf to sit on and sit under (also a good place to put any heavy items which might sink or squash them if on top of substrate and they tunnel underneath). A house big enough to build a large nest which is open underneath and sat on top of the substrate - so they can bury their hoards under their nest and build a big nest to keep warm. So the shoebox house Cypher mentioned is a good cheap solution. I also used to put a bendy bridge tunnel over the house door as that makes it darker inside and they tend to always use the house then as they like going through the tunnel into the dark. So a dark area to retreat to.
Diet is also important and some mixes aren't very good. A mix needs the right balance of protein, fat and fibre and all the vitamins plus low sugar or ideally sugar free. Tiny Friends Harry Hamster is a good basic option as it contains everything needed, has the right protein levels and the pieces are a good size for Syrians - they tend to eat all of it.
The Barney has narrow 7mm bars so I find you can fill the base with substrate to the brim and it doesn't fall out through the narrow bars
So a bit of time and patience now and in a couple of weeks he should be settled in and enjoying his home (and he may have even rearranged a few things - they sometimes cover up their toilet with bedding if they think it smells a bit or make mountains out of the substrate to hide a hoard or insulate the outside of a house).
Meanwhile taming and bonding can start by just talking gently to him through the bars and offering him the odd treat through the bars, so he gets used to your scent and the sound of your voice. You can then move onto offering a treat on the palm of your hand - just near one of the front doors. But best not to put your hand in the cage - it takes them some time to accept that and they get territorial in the cage and may run at your hand - outside the cage they behave differently.
If he shows signs of wanting to come out then the dry bathtub is a good placing for taming - they can run around and can't escape - putting a couple of toys in helps, and you can gradually get him used to being stroked on the back with one finger occasionally or a hand cupped under him and lifted slightly. When they feel confident about your touch they are easier to handle and won't try to "ping" (ie wriggle away and leap 2 feet in the air!). Always best to handle them low over something soft, or low in the bathtub in case they ping and hurt themselves.
A good way of transferring them from cage to bathtub is via a hamster ball if they hop into it -pop the lid on and carefully carry it to the bathtub and put it down and take the lid off. Then leave the ball in the tub as they sometimes hop back in to tell you when they've had enough.
Where is he nesting at the moment? If they can't find a house big enough they often build a big nest pile under a shelf, although sometimes they nest in a little house and then it gets messy and needs cleaning out a lot (the little plastic ones with a base sweat inside). So if he does that you might need to provide another house.