We Are Coming Father Abraham, Three Hundred Thousand More

It's more appropriate for the volunteers than the regulars, but here's another popular pro-Union song of the 1860s. "[We are Coming Father Abraham,] Three Hundred Thousand More" appears in Personal and Political Ballads (1864), though the earliest I've found it printed is 1862, in The American Crisis; Or, Trial and Triumph of Democracy (and in many music sheets that year).  Interestingly, this song is attributed to multiple people in the period publications; the lyrics are variously credited to "a volunteer's wife", and to William Cullen Bryant, while the Stephen Foster, George R. Poulton, and L.O. Emerson are all said to have composed the tune.  Wikipedia indicates that James S. Gibbons originally wrote it as a poem, which was published by Mr. Bryant and subsequently put to music by various people. An identical song, credited to "a volunteer" and increasing the number of recruits to 600,000, was also published the same year.

THREE HUNDRED THOUSAND MORE 

We are coming, Father Abraham, three hundred thousand more,
From Mississippi's winding stream, and from New England's shore;
We leave our ploughs and workshops, our wives and children dear,
With hearts too full for utterance, with but a silent tear;
We dare not look behind us, but steadfastly before--
We are coming Father Abraham--three hundred thousand more! 

If you look across the hill-tops that meet the Northern sky,
Long moving lines of rising dust your vision may descry;
And now the wind, an instant tears the cloudy veil aside,
And floats aloft our spangled flag in glory and in pride;
And bayonets in the sunlight gleam, and bands brave music pour--
We are coming, Father Abraham--three hundred thousand more! 

If you look up all our valleys, where the growing harvests shine,
You may see our sturdy farmer boys fast forming into line;
And children from their mothers' knees are pulling at the weeds,
And learning how to reap and sow, against their country's needs;
And a farewell group stands weeping at every cottage door--
We are coming, Father Abraham--three hundred thousand more!

You have called us, and we're coming, by Richmond's bloody tide
To lay us down for Freedom's sake, our brothers' bones beside;
Or from foul treason's savage grasp to wrench the murderous blade,
And in the face of foreign foes its fragments to parade.
Six hundred thousand loyal men and true have gone before--
We are coming, Father Abraham--three hundred thousand more!

Several recordings are available on-line, mostly under the title "We are Coming, Father Abraham/Abr'am".