I believe the current American Land Title Association Owner's Title Insurance Policy defines Insured as:
h. “Insured”: The Insured named in Item 1 of Schedule A and also the parties identified in Condition 2.b.
Section 2.b. reads:
b. This policy also insures:
i. anyone who receives Your Title by operation of law upon Your death;
ii. Your spouse who receives Your Title because of a dissolution of Your marriage;
iii. Your Estate Planning Entity to which You transfer Your Title after the Date of Policy;
iv. any beneficiary or distributee of Your Estate Planning Entity who receives Your Title;
v. anyone who receives Your Title by a transfer effective upon Your death as provided by law; or
vi. another Insured named in Item 1 of Schedule A.
So, if you, and perhaps your attorney, believes your LLC fits one of these definitions, it may be an Insured under the existing policy. You might also ask the title agent you're dealing with if they would be willing to issue an endorsement to the policy changing the name of the Insured to your LLC and if so, what would it cost.