One option I can think of is for you to create a dummy queue.
Insert an IF condition before the main menu, checking if someone is logged in on that dummy queue.
If statement is TRUE, then play a different set of menu options (this can be your "hidden menu")
If statement is FALSE, then play the normal menu options.
You can then ask your customer to use reskilling tool to assign the dummy queue to any agent logged in.
Basically, if someone is logged into the dummy queue, it will trigger the hidden menu. If no one's logged in to the dummy queue, then the usual call flow will be triggered.
I am not well versed in UCCX, but I use the same logic in UCCE for some of our sites.