The first thing to know about "The Tempest" is that, despite the name, it is not a version of "La Tempête". The latter is a different dance with different music and a much longer and more exotic pedigree.
"Tempest" is a mid-nineteenth-century American contra dance with a delightfully unusual formation, as shown at left in a (relatively) modern illustration from Ricky Holden's The Contra Dance Book (1956). The squares represent gentlemen and the circles, ladies. There are earlier illustrations, but this is the only one I've found that shows the dance in action.
Descriptions of the formation are very clear that the dancers line up couple facing couple, widely spaced, with the frequent suggestion of six to eight couples on each side, though there's no reason beyond the length of dancing time that there can't be more.
The dance is started off by the two couples at the head of each line peeling inward and going down the middle four abreast, as can be seen in the diagram. As those two couples progress down the set, each pair of couples reaches the top, waits out once, and then turns into the line of four to begin the dance in turn. This is the standard early progression style for country dances, as described in detail here. By the mid-nineteenth century, however, it was not unheard of for multiple couples to start simultaneously in a country dance. Boston musician-caller-author Elias Howe, whose books are, collectively, one of the major sources for "Tempest", was quite explicit about this. So it would not be unreasonable to start multiple lines of four simultaneously. The dance would continue until all couples have had a chance to rise to the top of a set, dance down it, and return to their original places.