Popup message after redirect OR message after user visits from a particular site

Popup message after redirect OR message after user visits from a
particular site

I'd like to show a popup welcome message AFTER a visitor has been
redirected from my old site to the new one.
I'm going to set up the old site with 301 redirects to the new content -
and only people coming from the old site should see the welcome message.
Users who access the new site directly should not see the message.
I know that I can set up a message before redirect on the old site - but I
currently only have access to my client's new site.
I can't find anything anywhere about doing something like this. Is it
possible?