Will a chin implant help move the part of the chin underneath the lower lip forward as well?

...you are wrong about your approach and judgement regarding your chin surgery; an osseous genioplasty is the golden standard, safe, practically painless and of swift recovery.You are right about your knowledge, your goals are only feasible with a genioplasty on the chin's bone.

Without an exam your question cannot be answered.  You will need to see an Oral/Maxillofacial surgeon to see if othognathic surgery, genioplasty or using an implant is best for your specific case. It should be noted that patients with severely weak jaws or abnormally functioning jaws may be better treated with other facial surgery (orthognathic surgery), although a chin augmentation may be an option at a later time. These patients should seek the advice of a plastic surgeon to determine which course is right for them. Here is some information to help you make this decision: Candidates for Chin Augmentation - Patients whose chin profiles are weak but have normally functioning jaws are candidates for Chin Augmentation. Many of these candidates have used injectable fillers in the past to strengthen the protrusion of their chins, but are looking for a permanent solution. If you are one of these patients, then you are a likely candidate for chin augmentation
#weakchin #chinimplant

Chin implants are only approximately 1 cm in height, and are placed directly over the lower edge of the mandible. Placing the implant  over the genio-labial sulcus will create unwanted pressure on the tooth roots and it is in the wrong location. For more information, diagrams of the implants and many before-and-after examples of chin implants, please see the link and the video below

An implant will  increase chin projection and improve contour up to the horizontal crease of the chin. The implant will not change the lower lip which extends from this crease up to the top of the lower lip.

Chin implants give a very  pleasing aesthetic result for patients with a chin too small for their face. As to what might be the right operation for you, there would be no way to comment without photos. 

 Hi, I have performed many Chin Augmentations using silastic chin implants for over 30 years.  When properly positioned, a silastic chin implant is designed with the purpose of adding forward projection to the chin to correct a weak chin.  When the  chin is weak, a facial imbalance is created making the nose appear larger, the mid face top heavy and the lower face look short that de-emphasizes the lips and allows early formation of a double chin.  Chin augmentation using a chin implant will add the necessary forward projection to the chin to create harmony and balance to the lower face. 
 I have found that placement of a silastic chin implant, through a small curved incision under the chin (also allows excess skin removal) to be very safe, quick, highly effective and far less invasive than a sliding genioplasty.  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.  
Hope this helps.