Here’s my take on this. Recovering data directly from an iPhone without a backup is a difficult task due to the way iOS handles security and data deletion.
I can see two potential paths, each with its own pros and cons. First, you could try third-party data recovery software. The main advantage is that it attempts a direct scan of your phone’s internal storage for recoverable data fragments. The downside is that success is far from guaranteed on modern, encrypted iPhones, and results can be mixed.
A more reliable, though indirect, option is to check your call history with your cellular provider. You can almost always log into your account on their website to see a detailed log of incoming and outgoing calls. This won’t restore the log to your device, but it should provide the list of unsaved numbers you need.