You'll need to find the contacts record on your other layout and sort them in the order you want.
In a script, you can use two basic steps, Go To Related Record and Sort to get what you want.
Your button then performs this script.
In the following link, you can skim a big part of it and focus just on the portions that deal with GTRR from a portal row. (The "bad things may happen warnings" aren't likely to be an issue with the task you describe.)
Go To Related Records is a very useful tool, but which is very poorly documented. To learn more about GTRR, click the following link:
The Complete Go To Related Record