If you’re allowed to add a name to a choleh then presumably the number of names you have isn’t a big issue
Please note that this procedure is known as “Shinui Shem” and not “Hosafat Shem”
This means changing a name, because in olden days, they simply changed the old single name to a new single name, when double names was not an option.
Nowadays when we anyways are utilizing double names, and are not on level to obliterate an already existing name, we simply add a name, albeit we still describe the procedure as changing the name.