"On" can have many, many different usages. The short answer is yes, most of the time you would want to use "to" along with "add on": "Add the spare tire on to the bicycle rack." And you might want to combine on and to into onto.
One exception would be if you wanted to use "on" as a preposition of place if you want to emphasize that you are adding something literally on top of something else: "Add the brick on the wall." (This scenario is rather contrived, though; if you wanted to emphasize the location you'd probably be even more specific: "Add the brick, on top of the wall," or use a different verb: "Put the brick on the wall.")
And this doesn't answer your present question, but in both cases, you can leave out "on": "Add the spare tire to the bicycle rack. Add the brick to the wall."
The fact is, the whole "Add ___ on to ___" construction isn't used very often. What we hear more often is a noun, in which something is an "add-on" to something else ("This expansion is an add-on to the original card game"), or a passive construction rather than active ("This book has an epilogue added onto the story"). The active construction "Add ___ on to ___" can be used as an extension of these practices: "Add the expansion pack on to the main game." You may have been wondering why we might stick with on to, two words, rather than onto: The only reason would be to preserve this connection with "add-on," in which "on" is part of a phrase along with "add." But you could also dodge all the issues by simply using "add" as its own verb, along with the preposition "to": "Add the expansion pack to the main game." I can't think of any scenario in which you would be forced to include the "on."