It sounds like he has lots of out of cage time which is good and they can sometimes be fine with a smaller cage if they have lots of out of cage time. But yes you can fit more enrichment in (variety, textures, etc) with more space in a cage and it becomes a real habitat for them. Having said that it isn't just the size of the cage. Some people get huge cages but the hamster is scared because there's a lot of open space and not much to do. As prey animals they tend to like overhead cover - eg a shelf or dark house or somehting hanging from above to sit in or under.
Changing cages can also be stressful but not if done properly
I've always let them have a play in the new cage the day before moving with it partly set up with bedding and any new items. Then moved them over the following day. The key is not to clean anything from the old cage when you move things over so it all still smells familiar (whereas our instinct is to want everything clean and new in a new cage) and to have the layout similar. They adjust in a few days if they've had time to familiarise and explore the new cage and it also has some of their scent in it.
If you are thinking of upgrading it can be good to have a bigger cage before they get old. When they get old and balance is an issue then everything needs to be at floor level so the bigger the floor area the more they can still enjoy their toys - all at floor level.
It depends how big your current cage is. If it's smaller than 75cm by 45cm I would say it's worth upgrading. If it's around 75 by 45 and he's happy then maybe not - especially as he has so much out of cage time.
I have one hamster in an 80 x 50 cage and he's happy in that - but if I wanted to give him that big labyrinth house it would need a bigger cage to fit in! The labyrinth house is in the Savic Plaza.
I actually do want to give him a labyrinth house as his current house doesn't quite work for him (he ends up sleeping in his toilet!) but at the moment he is happy in the 80 x 50 cage.
I do like doing the roof run though - it is more places to go - the hanging plastic things are rat sputniks. If hung next to a shelf they are easy to get into and all of ours like sitting in theirs and looking out at the world - I joined two together by tying a rat tube to the roof inside and they both enjoy their roof runs.
It's not the same as the smaller external tubes which are long runs - it's just one length of tube inside the cage - mainly to stop them jumping out of the other side of the sputnik.
Have a look on the "lets see your cages" thread - there are some amazing and fun set ups for ideas