Bird reaches around the collar?  (The following concept is shared by Dr. S. Medlin. )

To extend the width of a collar, select appropriately sized clear tape or cut strips of clear packaging tape. Fold sections of the tape to create a flap of tape adhering to itself with a free tab on both ends to adhere to the collar. (See illustration below.) Use overlapping tabs to complete each half of the avian collar, leaving a small gap where the two halves snap together. 

(1) Separate pieces of tape can be folded circumferentially over the tabs to join them into a single ridge or . . .

(2) left alone as individual overlapping flaps. 

The  extension can be short, medium, or wide. After placing the two halves together on the bird, add cross pieces of tape to complete the 360 degree extended rim.   You can trim the tape to a more pleasing circle or leave it square. Do not leave the sticky adhesive side of the tape exposed.

Yes, your bird will pick at the tape rim and you may need to reinforce it from time to time with more tape.  Most birds will adjust to this comfortably and stop trying to remove pieces of the tape rim. 

You are most welcome to share your personal techniques that you have found to be helpful. Send to < drghn@aol.com >.