I have been using a P2PKH address (i.e. 1Bxxx) as my paper cold storage since I started trading btc. I have only ever used used full-node bitcoin-qt to access this address. Also, I have only used a paper-copy of the private key-- I have never backed up the wallet.dat file. I simply manually import the private key into bitcoin-qt anytime I want to set it up on a new machine. As of right now, I have no machines currently running the node and no copies of any wallet.dat files I have ever used-- I only have the private key for the P2PKH address, as this was my understanding of how "paper cold storage" worked.

On May 5 of this year, using an up-to-date version of bitcoin-qt, I transferred 0.01 btc to an online vendor. From the blockchain explorer, it looks like the rest of my funds were transferred to an auto-generated P2SH address (3xxxx). I had not noticed this at the time, and have since wiped the computer from which the transaction originated. i.e. I don't have the wallet.dat, and I also don't have the the private key for the new P2SH address. Is it possible to retrieve these funds, when all I have is a paper copy of the private key for the original P2PKH address?