How to install 1/8" collet

When I got my XCP, I the 1/4" collet was installed in the nut, and the nut was mounted on the spindle.

I need to run some 1/8" shank and 1/16" shank bits, now.
I got a tiny collet along with the machine, and someone at Inventenables told me to simply push the 1/4" collet out, and snap the small collet in. Well, the small collet is so small it falls right thru the nut.

Am I to get a different nut for the spindle?

I run a Kress spindle. All of my collets have their own nut. They do make adapters which are sleeves that adapt the diameter of the 1/8th inch bit to the 1/4 inch collet. I wonder if that is what you have. Most people recommend using real collets and avoiding adapters.