MOST IMPORTANT ... talk to the H-1B company's immigration attorney.
Sure, it is possible to file for the H in April .... with a start date of 1 October 2013, or later.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.