Sounds like getting a cage another cage would be a good idea
Tall cages aren't really great for hamsters - and usually have smaller bases. Ideally something like the Hamster Heaven (80cm x 50cm and not too tall) - or larger. Maximum height about 45cm unless there is a full level. They do much better with more floor area and less height.
I know exactly what you mean about Syrians - I love them too - very handleable and yes they can be more cuddly or go to sleep on you.
You can often pick up second hand hamster heaven cages on gumtree or preloved as well.
It's taken me time to learn, but syrians do best with a few basics
1) A good sized floor area - at least 80cm by 50cm
2) A large house/nesting box at floor level (open underneath and sat on top of the substrate) - they like to bury hoards under their nest and it enables them to have normal habits like burying hoards under their nest or burrowing down a bit
3) A shelf or platform - they seem to like going on or under a shelf and it's a good place to put heavier items so they don't sink
4) At least 4 to 6" deep substrate
5) Rat sized toys and tunnels ideally (pringles tubes are good)
6) A 28cm syrian sized wheel
My first hamster was in a rotostak thing many years ago. I bought another rotostak when we got our Syrian about 7 years ago (without doing any research) and realised it was totally unsuitable. Our hamster got quite large and got stuck in the tubes and there was hardly any ventilation. So I googled and kept coming across this forum. Upgraded him to a hamster heaven sized cage and it made a huge difference - you could almost see him smile and his behaviour was much more confident. And he loved the bigger wheel.
Since then I've learned that the basics listed above are what really work for them. Most commercial cages are totally unsuitable. The tall ones have injury risks from falling (many hamsters end up injured that way) and the hamsters end up with funny habits due to not being able to pee at ground level if their house is too high - or they block up tubes.
One of the other advantages of the above is that although it takes a lot of substrate initially, you end up using a lot less in the long run - because it's not necessary to do big clean outs. The hamsters develop natural habits and the cage stays clean and dry - apart from their pee area (and sometimes the wheel). So most of us on here recommend "spot cleaning" - ie replacing the odd handful of substrate now and then if necessary. And not doing big clean outs. You can easily go 2 or 3 months with spot cleaning and emptying a litter tray weekly or bi weekly. And then only do partial cleans - eg when doing the substrate, don't clean anything else at the same time - do the wheel anotehr week and any other items another week again.
That way something always smells familiar and they don't get stressed by everything changing at once. I also replace about a third of the old substrate and spread it on top of the new so it smells familiar (the clean third - usually from the bottom of the cage).
A litter tray works wonders - they will use it if put in their pee corner/area.
Anyway over the years I've found it helps to really get a cage set up working before getting another hamster. They settle quicker.
The main thing with the partial cleaning is to avoid stress for the hamster - they do find cleanouts stressful - particularly if the nest or hoard is removed - and stress leads to them developing illnesses. I "prune" the hoard when I eventually do a substrate clean, as it can get too large and some of the dried food (although it keeps well hoarded) doesn't want to be around too long. It's only if the nest or hoard are pee'd on they need completely removing - and even then it's a good idea to leave a little bit of the old hoard or nest behind - and add new food to replace the hoard (in the same place) or they can get quite stressed about that as well.
With the strips of toilet paper for nesting, the nest stays clean and dry - you put out more now and then and they take it and refurbish the nest. Some of ours have actually cleaned out their own nest at times! If left. I've never needed to remove a nest unless pee'd in (which baby hamsters do sometimes but tend to grow out of it quickly). I've come down and found a pile of substrate at the cage door and half the substrate missing from the middle of the cage. So they've chucked out some stuff and dragged some new stuff in!
Our rescue syrian was the tidiest one! He was incredibly house proud (I used to say he was institutionalised lol). His pee corner was well away from his house and very neat.
Anyway. If you definitely want a syrian then thinking about a cage in advance would be a good idea. There are also lots of ideas for set ups on here on the "lets see your cages" thread. Some are huge! Most are average suitable set ups.
The Alaska cage used to be the basic, economical one recommended but it's not available since the pandemic.