Regardless of what triggering system you choose, the most wireless control you will ever have over the YN560 II is the command to "fire". The YN560 II has no capability for the power or zoom to be manually set via a wireless trigger. So your options for firing the YN560 II are either as an optical slave using the built-in optical trigger or to attach a wireless radio receiver to the flash that tells it when to fire. Radio has a few advantages over optical, especially when shooting outdoors under sunlight. You may or may not shoot in situations that require those advantages. Radio tends to have a greater range, doesn't require line-of-sight, can operate in very bright environments such as direct sunlight (that gives the optical system a tough time), and has the ability for more than one set of the same type to be used in proximity to one another without interfering with each other. (Think several press photographers all using Canon covering an event for multiple publishers. Or more than one shooter at a wedding.)
Since the only thing you're going to control on the YN560 II is the "fire" command you probably need to base your decision regarding triggers based on what works with the new flash(es) you buy.
For the YN560 III or YN560 IV the obvious answer is a YN560-TX transmitter that can communicate directly with the built in receivers of those flashes.
You then only need to add an RF-603/RF-603II/RF-605 receiver to the YN560 II to fire it via the radio in the YN560-TX. (Note: the Canon/Nikon specific models only differ in the "wake-up from standby" feature. Either can be used, minus the "wakeup" feature, with any model of camera that has a central pin on the hot shoe. But be sure to match your RF-603II to your YN560-TX in either Nikon or Canon flavors.)
Or you could choose to trigger the YN560 II in optical slave mode if the conditions you shoot in permit.
You might also consider the Godox TT600. Current compatible transmitter units include the Godox X1T-C and X1T-N (for TTL + manual control in the Canon and Nikon systems, respectively), as well as the XT-16, and FT-16 and Cells II trigger sets. Unfortunately, I can't seem to find a Godox receiver that allows a simple manual "fire" command via the hot shoe pin to a non-Godox flash without also paying for either Canon or Nikon TTL compatibility that you don't need.