Most sources I've read over the years as a student and teacher advocate doubling up on the letters should you run out.  Therefore, once you've assigned "z" to the end of a line, you would then assign an "aa" to the next line that introduces a new ending sound.  Looking at the opening lines of the poem, it is not difficult to see why one would quickly run out of letters when assigning rhyme scheme:

LISTEN, my children, and you shall hear A
Of the midnight ride of Paul Revere, A
On the eighteenth of April, in Seventy-five; B
Hardly a man is now alive B
Who remembers that famous day and year. A 5
He said to his friend, 'If the British march C
By land or sea from the town to-night, D
Hang a lantern aloft in the belfry arch C
Of the North Church tower as a signal light, D
One, if by land, and two, if by sea; E 10
And I on the opposite shore will be, E
Ready to ride and spread the alarm F
Through every Middlesex village and farm, F
For the country folk to be up and to arm.' F

When read aloud, one can hear the rhythm that makes the poem enjoyable to read; it is relatively fast paced, making it an ideal poem to read aloud to children.  Interestingly, although quite rhythmic, it's also somewhat irregular, and a quick look at the meter will show us why.  In some places, Longfellow employs the iamb, which is a rhythmic device of one unstressed followed by one stressed syllable (da-DUM) and the anapest, which is simply two unstressed syllables followed by a stressed (da-da-DUM)