WHY ONLY “HER PETTICOAT HOISTED UP ON HER HALF HITCH”? WHY NOT HER GOWN?

It may be that this part of the song dates back to the Elizabethan period when the word “petticoat” had a different meaning. according to a site on making women's garb of this period:

16th century English women generally wore a lightweight garment called a smock or shift as the first layer. This was of fine thin linen, the best of which came from the Netherlands hence “Holland smock, Holland handkerchief”—or silk for the rich— and was loose-fitting, often with sleeves and a collar attached. [in Scotland, sark]

A woman would wear over this smock a supportive garment, at that time called a petticoat.   [in Scotland, kirtle] The top part of the petticoat was a sleeveless bodice (‘bodies’ or “upperbodies’), that laced in front. The bodice of the petticoat was often stiffened with a layer of buckram, canvas, or fustian, flattening and supporting the breasts and pressing them upward as fashion of the time required.

“petticoats without bodies” meaning just the skirt.

The lower part of the petticoat played the role of providing extra warmth, adding shape and volume to the hips and skirt, and maintaining propriety by blurring the hip and leg outline, even when damp.

Some petticoat bodies were actually more of a long A-line dress without a waist. In that case the wearer would don a corset over the smock to provide breast support and give the proper line in front, and probably wear an overskirt and a belt to define the waist. Scots proverb “near is the kirtle, but nearer is the sark.”

Unless the wearer was well off, the petticoat would serve as the primary outer garment. Sometimes a woman who could afford it would make a stiff front piece with embroidery, which was pinned or sewn on over the front lacing to conceal it and give the illusion of a gown. Sometimes a stiffened ‘stomacher’, often of a contrasting color, was inserted between the shift and the petticoat bodice.   The bodice and skirt portion of a petticoat were often of different colors, and a wearer might make more than one bodice to attach to her skirt to change the look.

Those who could afford it wore another layer or two over the petticoat.

The gown was usually a garment composed of a fitted bodice with attached skirt.