Yes, given that you were already counted against the numerical "cap" on H-1Bs in 2013, until October 1, 2019 any employer will be able to take advantage of that fact and file a "cap-exempt" H-1B petition on your behalf. (That could be a good selling point on your part to new, prospective employers.) Note, however, that the years you spent and are spending in L-1B status will be counted against and subtracted from the 6 years maximum allowable under h-1Bs, until an employer corms and files a PERM on your behalf 365 days prior to your H-1B's "max out" period. You will then be able to perpetually extend your H-1B in 1and 3 year increments until you are finally able to obtain the green card under your employment-base category of your country of birth.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
If your H-1B was under the CAP yes you may use. If was CAP exempt you may use only for CAP exempt employer. You may only use if you have time remaining counting the L and H time already used. You need an employer to petition for you.
Attorney Robert Brown's (former INS Director, 1972-99) reply to your question is general in nature, and does not constitute legal advice as all facts are known to him. For specific advice or representation you should retain an attorney experienced in immigration law. Mr. Brown's reply on AVVO does not create an attorney/client relationship not constitute legal advice.
Partly used H1 can be used in future for balance of the period.
My answers are for general information only, NOT A legal advice because these are not tailored to your specific situation. Contact an Immigration Attorney if you need legal advice.