If you and your surgeon didn't feel a need before surgery to address this area, the swelling and fullness you see now under your chin should improve over time, though this will take several months to completely settle. I'm sure both you and your surgeon will agree that nothing was "added" to this area other than post-operative swelling and perhaps some fluid or blood. The latter can cause increased scar fibrosis and result in subcutaneous firmness and persistent bulging in this area, but can also be reduced with massage therapy, infrared treatments, and ultrasound (we offer these treatments as part of our post-operative care regimen in our office surgical center), and possibly steroid injections. If a collection of fluid or blood is identified early, this should be addressed via aspiration or re-operation, but this is not possible now--the fluid part is long gone and all that remains is swelling and/or scar tissue.
What is clear is that only time and resolution of the swelling component first, and then determining the scar fullness component, is needed before considering any kind of surgical revision. That way a proper and effective plan can be created. Rapid re-operation is something that patients (naturally, but incorrectly) want, and frankly, inexperienced and perhaps defensive surgeons fall prey to--premature re-operation is rarely good, and less likely to work out as everyone hopes. In fact, it often makes things worse (how do you think I know that?)
So, my best advice is to be patient for now, remain on good terms with your surgeon (not blaming or angry but understanding that he did his best and wants his best possible outcome for you as much as you do), and figure out a joint plan. I always do my utmost to make less-than-thrilled patients happy when they are thankful for what I have done, appreciative that I did not willingly wish their concern upon them, and are willing to do their part in covering OR and anesthesia costs as we discuss before their first operation.
Realize that the under-chin area is the most dependent area of the face and neck after surgery in these areas, and swelling, blood, and fluid, naturally tend to "go south" towards this region. Sleeping with your head elevated for the first three weeks after surgery and wearing an elastic shin strap for two weeks helps the tissues to adhere and heal evenly.
Neck bands are visible platysma muscle fibers that are visible before and sometime after facelift surgery. The exact procedure determines whether these are reduced, eliminated (rarely), or left alone, but complete elimination of these bands requires cutting the muscle--some surgeons do this fairly routinely; others decry this as causing a (different kind of) deformity. Partial division can provide some improvement, but would involve re-operation. Botox can offer non-operative relief of these bands, but need to be repeated every 4-6 months. Liposuction does nothing for muscle bands, but re-operation to address the residual fullness and the bands should be tailored to your requests and level of concern, but not before 6-12 months has passed. Best wishes!