Yup - I personally let them have an explore/play in it the day before moving over as have found it really helps them adjust quicker to the new cage - but some of that may depend on the personality of the hamster. It just means when you move them over it's already familiar and it's not so cold turkey.
So I fill the base of the new cage with substrate and put any new toys you have in there, plus maybe a familiar one from the old cage and let the hamster go in and explore. If they panic and want to come out again, then try a bit later. Sometimes on the second time they spend ages in there and explore everything.
When you do move them over then, you can leave the new substrate you've put in the new cage, but still move all the substrate across from the old cage and just spread it on top (cage is bigger so it will all fit) - press it down if necessary. And as Shannon says try and put things in a similar layout to as before, although that isn't always possible.
The most important thing is not to clean anything when you move them, even though it's tempting to have everything all clean and new! Just move everything over without cleaning things - move their nest across in their house too, and any hoard that's under it (if it's a really large hoard then prune it a bit). Then no big clean outs for the next too weeks.
With a nervous hamster I found that meant 2 or 3 days of skittishness rather than 10 days of total freak out! Even if they enjoy it and are happy it's still a bit strange.
You can play it by ear a bit though
I think it can help to at least introduce them to the new cage before moving them. You'll also need somewhere to put your hamster while you're moving things from old cage to new. If you have a pet carrier you can pop them in there. If not then maybe in the dry bathtub with a couple of toys but they will need someone in there as well to keep an eye or they'll get scared.