Build an extension on the standard Lead Conversion functionality

Build an extension on the standard Lead Conversion functionality

I am currently looking to create my own convert function and button.
The reason is that I have a connected custom module, which is  - at first connected to the lead, but all custom module records should be moved to the contact.

While creating and testing, I see that I miss a lot of standard convert functionality, like the interface allowing me selecting a existing Account. ( As per now , I keep on creating new Accounts in my custom script )

Is there an easy solution for this? How can I handle the fact that the records of my custom module should be transferred from the Lead to the contact?

It seems that writing my own custom function gives me more problems , instead of solutions.
An extension of the standard convert maybe??