Could it be the same one, as An old juvenile novel about a small group walking across the frozen Atlantic Ocean??
Time of the Great Freeze by Robert Silverberg.
Miles beneath the layer of ice that covered Earth in the New Ice Age of 2300 A.D., men survive in the subterranean cities they built to save themselves as the ice crept with killing cold over all living things. For three hundred years no one has seen the surface or communicated with any other city. Until now. Now the few scientific instruments that remain seem to indicate that the Ice Age may be ending; outside temperatures are reaching a level that may make life possible--though not easy--on the outside. But life in the undergournd (sic) cities is comfortable, and those few who are brave enough to be curious about the unknown frozen world above are suspect; troublemakers. A small party of these "troublemakers," led by Dr. Raymond Barnes, with a few scientists and others who think they might prefer freedom to safety, has been allowed to take the long-unused elevator up through the ice to the outside. But they go more as exiles than as a scientic (sic) expedition; they are not expected--and may not be allowed--to return.
It was written in 1964, which fits your time frame, and involves a group coming out of an underground city to see if it's safe after an Ice Age.
Search terms: juvenile novel teenagers emerging from an ice age (I don't know why I thought "teenagers" from your description...)