Being on L1,can i work for new employer(sponsoring my H1) before Oct.

Hi ,

My name is Ved , I am a software developer with 6 years of experience. 

At present I am on L1 visa working in united states , My L1 visa is valid till Jun2015 and I want to continue working in United states . I was searching for H1 sponsor  and got recruited in one company , they are ready to initiate my H1 visa , but they also want me to start working with them from April 2014 . 


now my understanding is , being on L1 visa I can not work for other employer , and i can not start working on H1 before October. 

so is there any way I could join new employer in April, who is ready to initiate my H1. 


Could you please suggest me options i should be following .