The simple, but architecturally repugnant, solution is to use a global shared variable.
Many people will tell you that this is a bad coding technique. They are right. But you aren't asking for advice on best practices.
Simply create such a variable in both the caller and the callee. Procedures that don't need it won't miss it.
One of your "normal" programs:
/* p1.p */
message "p1, I'm normal.".
run common.p.
Your "special" program:
/* p2.p */
define new global shared variable special as character no-undo.
message "p2, I'm special!".
run common.p.
message "special = " special.
The common program:
/* common.p */
define new global shared variable special as character no-undo.
message "common stuff...".
if program-name(2) = "p2.p" then special = "special value".
return.
You can define a NEW GLOBAL SHARED variable as many times as you like and you only get one copy of it. The "new" doesn't overwrite any existing variables (if you leave out GLOBAL it behaves differently).
You didn't ask for it and maybe you don't need it but the program-name(2) check peeks at the call stack to see if common.p was called by p2.p.
There are other, more complicated ways to do this but they all boil down to the same problem -- you're creating the basis for some very ugly coupling between your "special" program and the now no longer generic "common" program.