That laundry basket looks perfect for the job! With pip I had this contraption. It's a ferplast pod thing that is designed (cringe) to attach to a hamster cage as a place to go. You can attach tubes to it. So I had the pod in his cage with one 3 way tube attached. 3 way because the top part was capped off, but slowed him down shooting out of the tube because he used to stop and investigate the top exit!.
So I had some bedding in the pod and would put the odd treat in there. Pip liked to go in it and rummage around. So when I saw him in it I would put my hand over the tube to lift him out. The second he saw my hand enter the cage he would run for it, but the extra length of the pod meant my hand got to the end of the tube before he did!
Then I'd lift the whole contraption out and put it in the playpen. Eventually he just used to go in it because he liked it and associated it with coming out.
My attempt at handtaming was a cardboard box. When he'd had enough I used to put it on its; side int he playpen so he could run out, but your round laundry basket looks much better!
Anyway this was the contraption in his playpen - I don't have a photo of it in his cage, but it lived there.
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he loves to be stroked though and keeps still when I stroke him. I think it will get better when he’s older and not so full of energy 
last night when I went to scoop him up, he climbed onto my hand himself. But he hops off after a few seconds because he wants to carry on in his wheel. He loves running in his wheel 
My fault for posting my robo taxi!
i was frightened to hurt him. I think most people fail in hand taming because they fear of hurting these precious little fluff balls.
I am so glad I listened to Yorkies advice and very grateful
he punctuated two small holes in my thumb, but a horse bite is worse 
prints 
lots of enrichment