I have leads that are mapped to a stage in my deal pipeline - Stage is IDENTIFIED. 
When they reach a score they are converted to a CONTACT. 

I have mapped the Deal for Leads, but can't seem create the same fileds to map in CONTACTS. 

I know that this set up (lead having a deal) is unconventional, but this is how we want it set up so we can see all our CRM entries in deal pipelines. 

So, in leads we have a lookup picklist. How do I recreate the same picklist in CONTACTS and ACCOUNTS?

OR, shall I just create a new deal for these new converted contacts during the convert process and assign them the next stage? 

Hope I expained this well enough,