American Goldfinches exist in Ohio year-round, but they do migrate, so the ones you see at your feeders in the winter may not be the same ones that bring fledged young to your purple cone flowers in late July or early August. Compared to most songbirds, American Goldfinches nest later. Right now, the male AMGOs have fully changed to their bright yellow breeding plumage. In the winter, both males and females look nearly identical with a dull green color.
From the book "Birds of Ohio" by Bruce G. Peterjohn :
American Goldfinches migrate across Ohio each spring, but these movements are not particularly evident except along western Lake Erie. Their northward movements normally begin by April 10-20, peak between April 28 and May 18, and continue through May 25-30. During the first half of May, daily totals of 50-300 are noted along western Lake Erie and 1,000-1,500 have been estimated during sizable flights.
Their fall migration is normally less apparent. It produces totals of 30-60 daily at most localities, with occasional movements involving 100-200+. A few migrants are noted along Lake Erie by the first week of September, and they appear in most counties by September 20-October 5. The largest numbers are observed between October 10 and November 5. Their southward movements are largely terminated by mid-November.
In the month of May, if you easily want to see colorful migrating birds that wintered in Central and South America and the Caribbean, visit Magee Marsh, which is about 20 or 30 minutes east of Toledo. These are some of the birds I saw today at Magee (not my photos).
Black-throated Blue Warbler
Cape May Warbler
Black-throated Green Warbler
American Redstart Warbler
During the month of May, I also see and/or hear warblers in the oak trees around our West Toledo home. Just about any little woodlot in Toledo will contain migrating songbirds now. The numbers, however, will depend upon weather. These little guys migrate at night. The next warm, fair weather night with south or southwest winds will bring in another batch of migrating birds, which includes warblers, sparrows, flycatchers, thrushes, vireos, tanagers, grosbeaks, etc.
Spring Songbird Migration Summary