Many Oklahomans have too much income to qualify for Medicaid under state rules, but not enough to qualify for subsidies to help them buy private insurance in the federal marketplace.

“Many of the people we see are destitute,” Ms. Carmichael said. “But they fall into the gap, and there is little we can do to help them.”

In Oklahoma, parents with dependent children generally cannot obtain Medicaid if their household income is more than about half of the poverty level — higher than $9,240 for a three-person family. Adults generally cannot get Medicaid at all if they do not have dependent children and they are not elderly, disabled or pregnant.

In states that have expanded eligibility under the health law, Medicaid is available to people with annual incomes up to 138 percent of the poverty level, up to $27,724 for a three-person family and $16,242 for an individual. And states that have expanded Medicaid eligibility have generally had much broader gains in insurance coverage.

“Consumers come in, eager to see their coverage options,” said Jesus Frias, an insurance counselor at Variety Care. “We go through their case, we help them apply, and often we find that they are not eligible for Medicaid or subsidized coverage in the marketplace.”

James E. Cooksey, 54, visited the Health Department here last week and left with nine prescriptions for drugs to treat very high blood pressure and diabetes. Mr. Cooksey said his minimal income came from work as a Salvation Army bell-ringer in the holiday season and from odd jobs like mowing lawns. But he does not qualify for Medicaid because he lives alone and has no dependent children.