When I got my two budgies a bigger cage, I ”let them “find it” and choose it at their new home themselves. I simply left the doors of the new cage open all the time and put them near it while they were in their old cage. I put new toys in there and some of their branches and old toys. Along with some millet at the front door to entice them.
I then opened the door of their old cage (that they were currently still in) every day and let them have the option to explore it if they wanted. Oliver went into it day one and decided it was his new home and started hanging out in there, and eating and drinking. Sunshine on the other hand was more cautious and it took her about a week or two to be willing to go near it. Oliver spent a lot of time coaxing her over, because he wanted her to see it. It was the cutest thing. But I decided to wait until both my birds chose the new cage instead of their old - on their own terms. I also waited until I could see they had found the food and water bowls in the new cage, so I knew they were eating and drinking. And of course waited until they were very comfortable within it. The first few days they’d pop into it for a few minutes and then sunshine would leave quickly as she wasn’t sure about it. I waited until she spent a majority of her time (and Oliver too) in the new cage, and until they were favoring that one instead of the old one. Then, after all that I took away the old cage and they were happy and enjoying their new one. And they don’t seem to miss the old one at all.
I would recommend doing something similar if you’d like. Yes, it will take more time then just plopping them in the new cage, but it’s more exciting and less stressful for them if they can explore it on their own time and leave if they get nervous. I believe it’s better on them too as they are “picking” their new home. Good luck