J1 expires before October...can H1B be processed

I am at present on J1 visa with a University, have come to US for an exchange semester.This being the last semester of the overall program, I could also get the degree from my home university.

A company wanted to hire me here in US, and the University could extend my J1 visa until first week of April ( that was the maximum possible extension). 

 

I wanted to check with you- what other Visa options do I have to work for this company here in US beyond April first week ( when my J1 expires). My company can file H1B, but with that I can start my work only in October, and I will be out of status from April to October. Would much appreciate your advice. Many thanks.

 

P.S: I am not suject to  2 yrs home country  requirement.