Hello. Yes he does need more space
It's a good time to upgrade him as you've had him 3 weeks or so and the current 10 gallon could be considered as a starter cage.
A hamster needs about 80cm by 50cm of floorspace in a cage/tank - more if you can. Two smaller cages attached doesn't count to the floor area. The idea being they can have enough enrichment in the cage to have normal behaviours like digging, foraging, exploring, different areas for sitting, washing, hoarding etc. They also need a house or hide that is dark inside to retreat to and big enough to build a large nest in (especially in winter) and a shelf of some kind for some overhead cover so they can sit under it - and they like sitting on a shelf as well - somewhere to go!
Barred cages are great as they have plenty of ventilation and it's easier to interact with hamsters. I prefer them to tanks. Tanks are easier to get cheaper in the US though I believe - Petco has $1 per gallon sales regularly and a 40 gallon breeder tank is the usual choice for a Syrian (teddybear) Hamster. They are basically ground dwellers and actually need more floorspace than gerbils probably.
A tank topper isn't a great idea unless the floor area of the tank is bigger - they do much better with more floor area and less height - they aren't great climbers because they don't see well and tend to only climb to see if they can escape out the top! When the do climb they're not good at getting down again and tend to just drop - so height can be dangerous as they can get injuries from falling from a height. With tall cages you need really deep bedding and possibly a full level to reduce the height and make an extra floor.
Good cages are hard to find in the US - that are big enough - there have only been a couple available for some time - both over $100 - the Lixit Mickey 2XL on Amazon and the Marchioro Goran 82 (the Goran used to be sold by Pet Mountain but always seems to be out of stock so may no longer be available.
However - I did come across a new cage on Amazon in the US which seems very popular and is a good size and good price. So I think that would be a good option:
Here it is!
https://www.amazon.com/dp/B074FH47MQ...525073471&sr=1
Other options would be - a 40 gallon breeder tank, a detolf or making a bin cage. A good bin for a bin cage is easily available at this time of year - the Iris Christmas tree bin
https://www.walmart.com/ip/IRIS-Holi...d-Red/48037027
It is half the price of the cage but involves some diy. You either need to cut out panels with a hot knife and attach mesh for ventilation with cable ties or bolts - or another option I've seen is to just replace the lid with metal baking trays - see photo below - but the grid trays may be expensive or hard to find
See first two photos on this link
https://www.google.co.uk/search?q=ir...w=1138&bih=503
The detolf option is an Ikea glass cabinet, tipped on its side to make a tank - with the door left off. Also involves some diy as it needs a lid making (most people make two half size lids so they're easy to remove) out of wood strips and mesh. That is if you have the space for a detolf. People often stand them on Kallax units with storage underneath which looks nice, but if you don't have Kallax units it adds to the cost. It used to be a cheap option for a good size hamster habitat, and perhaps more manageable than a 40 gallon breeder tank. But as it now costs $60 on its own, the cage at $69 looks like a good buy!
https://www.ikea.com/us/en/catalog/products/10119206/
Here are some examples of detolf set ups
https://www.google.co.uk/search?q=Ha...w=1138&bih=503
Personally I would go with the cage