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While checking my mail every night, I'm getting email notifications about new responses in threads, I posted in. So, I go there, then log out.
When I'm done with my mail, I'd like to come back to this forum, and check all posts since last night, not from my last login.
For what it is worth:
I believe I accomplish this by doing your steps in reverse order.
That is - I come to the site and click on Show unread since last visit. Then I check email, etc.
I can't see an option like that, or it doesn't exist?
I don't think so, but I'll be interested to see what other responses/methods you get.

By the way, if you haven't found the 'old' search function for the forums, it is here:
http://www.computerhope.com/forum/index.php?action=searchOften, I like it better than the newer search method.
The old search does allow for limiting searches by number of days. And allows you to search for posts by all users, using the asterisk. However, it does not allow you leave the Search for: box empty, to search for all posts. If it did, this would probably do exactly what you want. Then you could set it to search for all posts, from all users, for one day back.