You could invent a nice rule on how to generate a new private key from your private key.

Example.
You could take your private key an multiply it by any number (for example: 36453278)  to get the next private key then multiply that to get the next one and so on.  You can do the multiply as many times as you want to. to generate as many addresses as you need.

Just remember that we are dealing with modular arithmetic, so you need to take a mod from the result, so that it wont get bigger than the maximum allowed value.

eg:

(privkey1*36453278) mod 115792089237316195423570985008687907853269984665640564039457584007908834671663

That is basically what HD wallets do to generate many private keys from one. Except, they use a bit more complicated algorithm and sha256 hash on each result so that even a small change will create a completely different private key.

The advantage of this kind of approach is that you only need to backup your original private key and the multiplier, and not a huge list of different private keys.