Get reference to object in Set

I think this question has already been asked, but I have failed to work
out a good search term to find the previous discussion.

Suppose I have a Set<MyType> mySet and a MyType myObject.
mySet.contains(myObject) is true. I need a reference to the equal object
that is already in mySet.

Currently, I do this by maintaining a Map<MyType,MyType>. When the get
call returns null I do a myMap.put(myObject, myObject), so that future
get calls with equal objects will return the actual reference.

Is there a better way to handle this? Any way to make a Set return a
reference to the object it contains?