Ask them if they might have any leads or advice for you on finding an entry-level position. You may even want to ask if they'd be willing to meet with you one more time and critique your resume so that it might look more interesting to other people in the field. And finally , follow-up. Stay in touch, especially if the person expressed interest in your progress. If given a referral which turned out to be a gold mine of information, drop a note to the person who made the referral.
People appreciate knowing when they have been helpful.
Later, when you do enter the field and accept a promising position, a follow-up thank you would be polite, as well as wise.