What is the best word for a "nearby time"? Proximate?
Solution 1:
I would recommend the use of adjacent to suggest a time(frame) close to your original appointment:
a : not distant : nearby
b : having a common endpoint or border
c : immediately preceding or following
(See the "full definition" section at http://www.merriam-webster.com/dictionary/adjacent)
"Are you available any time adjacent to our original meeting?"
This should (hopefully) satisfy the comments that the word you're seeking should suggest either a time before or after the original appointment.