It can be a long process.
Filing for an H before getting a decision on the L is not a good idea.
What did the company's immigration lawyer say about this idea?
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.
The same employer or another employer can file an H1B petition for you, but as you are probably aware the filing will be constrained by the visa cap.
There are also issues of being granted change or extension of status to H1B if an intervening petition is still pending with the USCIS, so the question of whether an H1B petition can be filed is separate from the question of whether it is desirable. The pending L1B petition can easily take several months for a decision, esp. if there is an RFE. Please consult with an immigration lawyer.