Im new to contract work and working on resume for full time jobs. Add client or employer in resume ?

Hi folks

I got laid off last year and found work in contract with a consultant and working for a client for a long term contract. The contract hasn't converted to full time yet and im not waiting any longer.

Should I add that im working for the client as a contractor or just add the consultant company im working for. My client is a big fortune 100 company and employer is a big local name.

I do want to get an interview but not sure what to do in my resume.

However how do you present all this in interviews without it being complicated or getting caught in a lie later on. Especially when they ask why are you leaving the current place which i know is going to come up.

Im wondering if I should just add my employer and be done with it..m Any help from anyone in the same boat..would be great..