Yes, your suspicion is correct. Once man had boats (no later than 40,000 years ago) and the ability to live in the arctic, the island chains strung across the Bering Straight could not have been a significant barrier. There are native peoples who traverse it regularly today using native methods.
As for evidence, archeologically we know about the Thule people (ancestors of today's Inuit), whose culture is first found in the Chukchi Peninsula (west side of the straight) and the Bering islands around 200BC, and spread eastward across the straight and clear to the Atlantic Coast of North America.
Linguistically there appear to be no less than 3 waves of immigration that occurred across the straight (4 if you count the Inuit). Genetic studies show at least 2 (3 if you count Inuit), with the Na'Dene people almost certainly having come over after the land bridge submerged 15,000 years ago. Thus boats would have been required.
Genetic flow over Beringia
Present-day distribution of Na'Dene
For these reasons, very few scholars still argue that the only immigration from Asia to the Americas happened over an Ice Age land bridge.