Getting a supplemental increase after the fact is more difficult than hammering stuff out beforehand... you can't reinspect damages that have been fixed already and thus that opens the door to the insurance saying the repair wasn't necessary.
While others may have had more success in after-the-fact supplementals than I, my experience has led me to tell adjusters that nothing is getting done until the check's cut.
I guess it depends on the insurance company. Had this done twice at my house. With Aetna and with Safeco. Both paid the actual costs. Originally they take off for depreciation value, but add it back when you submit bills to show real costs. I also do work and submit extras to the insurance companies for about 18 years. vEry rarely are they not willing to pay what can be substantiated damages.
Most insurance agencies will pay the difference after you submit the bill from a contractor. Possible they may ask for two other quotes also. If you do the work yourself, they will only pay up to what the adjuster claims.