It’s a rare spelling variation. The OED says:
taut, adj. Forms: … β. 1600s– taught (now nonstandard)
and gives citations showing that this form remains current:
1834 F. Marryat Peter Simple II. xi. 174 The yards carefully squared, and the ropes hauled taught.
1948 Trans. & Proc. Amer. Philol. Assoc. 79 48 When the lines were drawn taught the sail was relatively flat.
1997 M. L. Frankel Cruising Gulags 175 Its telephone system..is marginally better than two tomato cans separated by a taught string.
This spelling variation carries over to the adverb, for although the OED does not record the spelling “taughtly”, it is easy to find many examples via Google Books:
1979 M. Langley Nandi of Kenya 43 It is made by drawing a piece of goatskin taughtly over one end of a small wooden barrel
1987 E. C. Apfelstadt Later Sculpture of Antonio Rossellino 161 The cast of characters is the same, with two taughtly drawn shepherds
2012 C. Vaporis Voices of Early Modern Japan 170 The noise was made by a thin shred of bamboo stretched taughtly across the top of the kite and played upon by the wind.
So in context in ‘The Jackson Killer’, “taughtly” means “in a taut manner”, where “taut” can be understood in the sense:
taut, adj. 5.c. Neatly or smartly turned out or presented.
This agrees with the description “smooth, well groomed” in the story.
Update
There was some skepticism in the comments as to whether the evidence presented above really shows that “taught” is a variation and not a mistake. Could every one of the examples listed be an accident?
I think that this kind of criticism is less tenable once you have looked at books that use only the spelling “taught”, and use it multiple times, showing that it was not only the preferred spelling of the author, but also acceptable to the editor and proof-reader too. Here are a couple of examples:
Taught (tight). The string or twine is pulled “taught” when a loop is made
Pull both strings taught taking care that the knot bites and does not slip
when the spoole is removed, draw the needle taught
loosened sufficiently to enable the workman to draw the last loop up taught
now let go the left thumb, pull the knot taught above the knot of the mesh
pulling towards you sufficiently to keep the margins taught
When you fling the twine round, and until you pull the knot taught
make a loop as usual, only just before you draw the twine taught
Charles Bathurst (1837). Notes on Nets; or, the Quincunx Practically Considered. London: J. Van Voorst.
Pull the masking tape fairly taught
This time use the canvas pliers to grip the canvas and pull it taught
Carry out the same process on the fourth side, using the pliers for a good taught finish
Test the tension of the canvas. It should be taught but also give slightly
light tapping of the canvas will pull it taught again
Heather Schofield (1979). Flower Painting Techniques. London: B. T. Batsford.
