Try these guys http://www.lieferwagen-mieten.ch/


Cheap and cheerful, don't expect a 2019 Mercedes but they do the job and you can drop off out-of-hours at no extra cost.

Given Sharoo a go, which on face value seems to be exactly what I am looking for. Not used them yet just made the booking but a great find!

check out at this one - it's a local garage just out of Winterthur, open on Saturday - you probably need to book for 2 days till monday morning to return it.

https://www.lord-garage.ch/miet-lieferwagen/

It all depends how well do you prepare the move and how much distance you have to cover.
If your household is not too big, should be done in half a day(if the new flat ist not too far).
You write the distance is 10km, so the question is how hany rooms you have to relocate ?