The at-fault driver's insurer paid my claim but not the full repair cost. Can I get the rest?

Your question does not name a state, and the rules differ by state. This answer uses pages from the Texas Department of Insurance (TDI) as an example. TDI says the other driver's insurance should pay for your car repairs, medical bills, and a rental car, but it might not pay all of your costs [1]. TDI lists "Say their driver doesn’t have enough insurance to pay all your costs" as something that company might do [1]. You do not have a contract with that company, so you do not have the same options as with your own company [1]. If it says it will not pay, TDI says to ask for its reason in detail and in writing [1].

TDI says that if you disagree with the adjuster's estimate or the amount the company is offering, you should tell the insurance company why. You might be able to work things out by talking with the company or the adjuster [2]. TDI says appraisal can be used only for disputes with your own insurance company, not with another person's insurance company [2]. Another path is a claim with your own insurer. For that you must have collision coverage, and you will have to pay a deductible [3].

You can also complain to TDI. The complaint must be in writing. TDI says it cannot decide who was at fault in an accident or determine damage amounts [2]. If the company will not pay for your damage, TDI says your next option is to get legal help [4]. For a claim of less than $20,000, TDI says you can use Justice Court, a special court for small-claims disputes. You do not need an attorney, but you have to pay a filing fee and other court costs in advance [2]. The TDI help number is 800-252-3439 [4]. The sources do not say whether you should sign anything the insurer sends you, or how to ask for more money after a payment.

Open the TDI page "Accident not your fault?" and read "Issues that can happen when filing a claim" and "Get help". Then open the TDI Auto insurance guide and read "Resolving problems". The sources do not say how another state handles this, and they do not say whether a release, a supplement request, or a second payment is allowed — "Don't know yet".