First, see this quote from Modify Unique Constraints:
To modify a UNIQUE constraint using Transact-SQL, you must first delete the existing UNIQUE constraint and then re-create it with the new definition.
This constraint (totally pun intended) extends to all sorts of constraints and all sorts of relational databases, by the way - NOT just unique constraints as the quote suggests, and not just SQL Server. You just can't modify existing constraints in place.
So, you can drop the constraint using this:
ALTER TABLE myTable
DROP CONSTRAINT myConstraint;
Once the constraint has been dropped, add the unique constraint with the additional column or whatever you wanted to do:
ALTER TABLE myTable
ADD CONSTRAINT myConstraint UNIQUE (column1, column2);