Oh..you are working on contract. So it could be that you have OPT from a contractor which could be e-verified and you have extended your OPT. If this is the case then talk to the company and ask them to file for H1B with start date of Feb (when your contract ends). For that, accept the employment offer from the company and ask them to file H1B. You can continue working with the contractor until your H1B start date. And, I think the company need not be e-verified for you to file H1B as you will not be working as their permanent employee until you are granted the H1B.  Just make sure that they do the premium processing for you to get H1B by Feb 1st.
 - As per USCIS website there is still 11,100 quota left for regular H1B and 300 for US Master Degree H1B.

This is just my suggestion with what I have understood the process. Your lawyer who could be preparing your H1B could answer you better.