One thing you can try is to build specialised subagents to help tease out more thinking when you're stuck with a model with limited reasoning budget as all the codex sub models seem to be (compared to API) --- but I have no firm, hard evidence of this so treat it as guesswork on my part.

Let me explain.

Creat a subagent that is told to just mull about a subject. This can be a simple file.

Have the agent expound on it with its own thoughts and additions.

Now rinse and repeat: give the file to another blank 'reasoning' subagent; then another; and so on.

This will eventually reach a point where they are unlikely to add more real value.

I believe they've locked down temperature and other sampling parameters so you cannot tweak these to get it to consider more unusual approaches.

The absolute best approach is a second model that is of a different family. Try Deepseek, GLM, Qwen, etc. and have it add its own thoughts. As a bonus, these models do not hide their reasoning. Put that in a file also.