Have you tried talking to the accomodation management? Ideally you'd want to move directly from one to the other, not having to worry about intermediate storage. I'd try and get them to move one tenancy a day (although should you not have queried this when you organised the tenancy for the second property?)
Anyway, if they're really arsey about it you may have to look into alternative single night accomodation (hotel, hostel if you have a location for your belongings or not very many of them, with a mate) and then move in the next day.