But on balance I think I would move him to the new cage with perhaps 6" of bedding to start with as the sooner he is in a bigger cage without those tubes, the better it will be for him generally. Then you can observe him and see if the patch clears up or not. Although he might love using the tubes, they are a bit small for him really, especially the curved ends at the bottom. I did have a hamster get stuck in tubes once! Hamsters can be up to 7cm wide if they have full pouches (and our hamster had grown particularly large). Wouldn't want you to waste money on the new one if it's not quite right either though, hence asking which one it is
If it doesn't clear up after he's been in his new cage for a few weeks, or starts to get worse, then it's probably mites and it would need a vet visit for some medication. And if mites are diagnosed then you would need to clean out the new cage and throw the bedding and replace it all (hence suggesting only 6" at first and waiting a few weeks as he'd be due a bit of a clean by then anyway).
When moving them to a new cage it's best not to clean anything. The opposite of what you'd think really. But it's important to avoid stress for the hamster and the way to do that is keep everything familiar smelling - his current cage contents will have his scent on. So for example you could fill the new cage with new bedding and then keep his old bedding and spread it on top (or at least spread it on top near a house area where he'd sleep). And move across any other items you're going to keep/re-use without cleaning them. Plus adding any new items.
It would be a good idea to have a large house, that's dark inside, in his new enclosure, so he has somewhere he feels secure to make a nesting area as he may be a bit freaked after being in the small cage. That doesn't have to be expensive - a shoebox house is fine (I'll mention that further down this post). It's also a good idea to have a platform in the cage as well - maybe opposite end to the house. So he has plenty of overhead cover and doesn't feel too exposed. They often like to sit under a platform and also to climb on it as somewhere to go. Plus a platform is a good place to put heavier items or ceramic items that might sink in the substrate
Shoebox house: You just cut the base out of a shoebox, keep the lid as a lift-off roof, and cut a hole for a door. The best place for the door is on one of the long sides at one end of it - so the other end is darker inside. If you have a bendy stick bridge, you could put that over the door too - as that makes it even darker inside (light doesn't go round corners) and also provides a ramp onto the roof of the house.
Hamsters like tunnels generally but I wouldn't move the plastic tubes to the new cage as they're a bit small. They do better with larger, rat sized tunnels at floor level really. Pringles tubes make great tunnels to put on the substrate, and if you buy one thing, a large cork log makes a good tunnel and is also something for them to climb over and an interesting texture. He'll also need an 11" or 12" wheel.
My first cage was an awful small thing with tubes leading to different levels. I also thought the hamster loved the tubes at first, but then I realised he had no option but to use them! As it was the only way to reach the levels. There were issues within the first two weeks, which was when I did some research and upgraded him to something suitable.