Here's what I believe to be the best way to solve it.

a = Alice, m= Mad (Hatter), h = Hare (March), d = Dormouse before the move;
A = Alice, M= Mad (Hatter), H = Hare (March), D = Dormouse after the move;.

Start by drawing the table and putting eight chairs around it.  Don't worry about directions -- save that for last.  Just stick Alice in one of the N, S, E or W chairs.  The other three people start in the chairs with the question marks, but we don't know which:

         a
         -

       -   -
     ?       ?
     -       -

       -   -
         ?
         -
Okay.  1 = "Alice moves to the seat directly between the Mad Hatter and the March Hare".  So after this move, the setup is one of these two configurations:

         a                        a
         -                        -

       -   -                    -   -
     ?       ?                ?       ?
     -       -                -       -
       A                            A
       -   -                    -   -
         ?                        ?
         -                        -
 

2 = "The Mad Hatter moves to the seat directly between the Dormouse and the March Hare".  In the first configuration, this means that the leftmost person moves between the bottommost person and the rightmost person; In the second configuration, this means that the rightmost person moves between the bottommost person and the leftmost person.  So first Alice filled the gap between the bottommost person and one of the side people, and now the Hatter is moving to a gap between the bottommost person and a side person.  Both moves involve the bottommost person. And, if you read the steps, you'll notice that both moves involve the March Hare.  So the March Hare must be the bottommost person.  (This leap is tricky to explain, but some thought should make it clear.)  Now we have one of these two arrangements:

         a                        a
         -                        -

       -   -                    -   -
     m       d                d       m
     -       -                -       -
       A   M                    M   A
       -   -                    -   -
         h                        h
         -                        -

3 = "the Dormouse moves to the next seat clockwise from the chair the Mad Hatter just vacated;".  Look at the two configurations above.  Notice that in the second configuration, the "next seat clockwise from the chair the Mad Hatter just vacated" is already filled, by Alice.  So it has to be the first configuration.

         a
         -
       D
       -   -
     m       d
     -       -
       A   M
       -   -
         h
         -

And since " the order of the party, going clockwise around the table, hasn't changed" after the move, you know that the Hare most go to the vacant chair -- which, according to the last stipulation, is the SE chair.  Now that you know where everyone was and is, you can go around the table -- starting from the chair you know is the SE chair -- and label all the directions:

        (E)
  (NE)   a   (SE)
         -
       D   H
       -   -
 (N) m       d (S)
     -       -
       A   M
       -   -
  (NW)   h   (SW)
         -
        (W)