My car was totaled and other person’s insurance will not cover all costs

Let's think this through. Unless the other driver was at fault, his insurance company will not cover anything. If both drivers were somewhat at fault only the driver less at fault can collect (in a few states -- but not NY -- any fault by a driver precludes recovery) but the less at fault driver only recovers a proportionate share of the damages.

The the issue is what do you mean by ALL costs? Renting a stretch limo with a chauffeur would not be covered. The wages you lose while getting the car fixed are not covered. It all depends on what you mean by "all costs."

If you have collision coverage, let your carrier deal with it.  You'll bear the deductible up front, but when your carrier goes to collect against the other driver's carrier, if the other driver was primarily at fault it is likely you'll get most if not all of the deductible back.

If you do NOT have collision, then you can take the value offered by the other driver's company, or try to demonstrate why the value is inappropriately low.  It's often a matter of negotiation. You also could say I will repair the car on my own, so pay me what you'd offered, less the salvage value they'd get for the car.  (When a car is totalled, it is similar to selling it to the insurance company; they take title and sell it to a salvage dealer.)

If none of these approaches work, you have the right to sue the other driver, perhaps in small claims court. Of course, you'd have to establish (1) that the other driver was at fault (2) you were not primarily at fault or contributorily negligent, and (3) the fair market value of the car that was destroyed. If you haven't repaired the car it is usually difficult to prove the cost to repair the car as unless the mechanic is there at trial, what he told you is "hearsay." If you have repaired the car, the paid repair bill often accepted as evidence of the cost to repair the car. 

Also, if you sue it is possible you could lose or get less than the other driver's carrier is now offering. On the other hand, to avoid having to defend the case, the insurance company on other side may up its offer.  It's a 50-50 shot as they also may take their offer from the table.