It depends on whether the trust is revocable or irrevocable. In the case of a revocable trust, coverage can depend on the number of trustees and the number of qualified beneficiaries (qualified beneficiaries include natural persons, charitable organizations and other non-profit entities recognized as such under the Internal Revenue Code). The same revocable trust insurance coverage rules apply to deposit accounts owned by one or more individuals in the form of Totten trusts (in trust for qualified beneficiaries) or with POD designations naming qualified beneficiaries.
You asked whether an account owned by a trust would be covered by more FDIC insurance, and the answer will also depend on what type of account it is being compared with. For example, if Harry and Sally own a deposit account as joint tenants with rights of survivorship, and it's the only jointly-owned account either of them has with Bank A, that account would be covered up to a balance of twice the SMDIA, or $500,00 under current rules. If the same account were held jointly by Harry and Sally and three nephews are identified as POD beneficiaries, and neither Harry nor Sally has any other revocable trust accounts with Bank A, the account would be covered up to two times the SMDIA (two owners) times three (the number of qualified beneficiaries), for a total of $1,500,000. Harry and Sally would each be entitled to revocable trust coverage of $750,000.
As for irrevocable trust agreements, funds representing the non-contingent trust interest(s) of a beneficiary deposited into one or more deposit accounts established pursuant to one or more irrevocable trust agreements created by the same settlor(s) or grantor(s) shall be added together and insured up to the SMDIA in the aggregate. Rather than try to decipher what that means, I suggest discussing the particulars of a situation involving one or more irrevocable trust agreement directly with the FDIC.