Travel US without LCA

Thank you all for the wonderful response here, been really helpful.

My situation seems unique and I could not find something already posted. Hence...

In Oct 2013, while in US, my employer did COS from F1 to H1 and I was working with him until Mar 2014. Quit my job and returned to India for various reasons. Now am planning to step into US for another job. My current H1 visa is stamped till Oct 2014. One employer suggested that I [b]travel to US immediately based on an Invitation letter[/b] provided by him. He assures me there would be no issues at Port-of-entry. Finding a job would be faster considering my skill-sets according to him. My questions are

1. Can I enter the US with my old LCA (which is still not revoked) though am not working for the same employer?

2. Would only an Invitation letter suffice for entry to US? My understanding is I would still need a LCA filed/approved by new employer.

Please do help me out on the right direction. I do not want to be deported and jeopardize my future entry to US.