Offer a new site before we have seen the old one. In a good half of cases the existing site is fine and needs updates, access and two corrections, not a rebuild.
Transfer the domain before it is clear who owns it. If it is registered to the previous provider that is a separate and sometimes awkward conversation. We settle it first anyway.
And delete content that looks dated. Old pages with rankings are worth more than new ones without. What goes is decided after looking at the numbers, not at the appearance.