Yes, hats of the "stovepipe" form were still fashion in the early 1860s. Worn both formally and in "morning dress" (basically business attire) by many.
In George Caleb Bingham's 1854 election painting, you can see top hats yet about, though the more comfortable slouch hat was most common:
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By the 1860s, the tall hats were still fashion..
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Cairo, Illinois post office, 1861, at least one chap in a stove-pipe at right:
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As mentioned above, they were not part of the period "military fashion" and not universally worn among the citizenry, but some were employed by militia, Confederates, etc. where necessary.
Here's a group of Charlestonians posing with a field piece in 1861, in their citizen's dress, some including top hats. Very likely militia.
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Here's a wartime Indiana Home Guard:
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When Company A, 6th Indiana Regiment of Volunteers formed in April, 1861, before they received any uniforms, the fellows posed in their civilian dress:
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Such hats are not the best for active military service, but considering the poverty of suitable clothing among the Confederates, they might be put to use in a pinch as evidenced by some period sketches.
Edwin Forbes sketched Confederate Prisoners taken at the Battle of Chancellorsville, VA May 3, 1863, showing some...
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After their raid into Ohio and Indiana, General John Hunt Morgan's Confederate cavalry were described as rather un-uniform, having appropriated a wide variety of clothing along their long ride by the time they were imprisoned at Camp Morton:
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An from a Waud sketch of some of the more destitute prisoners collected from Lee's army at the siege of Petersburg in 1864:
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Brigadier General Joseph Finegan, commanding Lee's Florida Brigade in the last year of the war, was noted as customarily wearing his citizen's coat and top hat and waving his cane at the Battle of Cold Harbor in June, 1864.
When Lee's army marched into Pennsylvania in June, 1863, the militia of the State was called forth to defend Harrisburg:
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At the battle of Gettysburg in 1863, local citizen John Burns joined in the battle in his modest stove-pipe:
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At President Lincoln's second inauguration in March, 1865, there was still a top hat or two to be seen among some gents in the crowd:
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And in 1866 they still might be seen among some in Congress:
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