Wallet Recovery Help
Unfortunately, the method used to generate new keys at the time was to generate them randomly and store them in the wallet.dat file. So if you don't have a backup of the file after the address was used, it is not guaranteed that your backups will have the necessary key.
I would suggesting dumping the wallet.dat file to see all the private keys contained within it. Either it has the key for the address the funds were sent to and you are in luck or it doesn't and you are out of luck.
One easy way to do this is to use the dumpwallet command. There are also tools available to parse a wallet file and extract all the keys in it.