This could be a problem on a free iCloud account with almost no storage. If the account has room, though, then it will keep multiple backups. I’ve had to create a new install, then go back and choose an older backup to restore, and it works fine.

On a free account, there may not be enough room for two backups, so the older one may be overwritten. If the person only has the free account, they could log into iCloud from any browser, upgrade to the 99 cent plan, go through this process, then downgrade again later if they want.