Grasp the top edge of the skirt fabric with pliers and pull gently, freeing the staples or tacks from the hassock. Continue around the stool until you remove the skirt. Remove the piping and the top piece of upholstery in the same manner. Label the pieces and their locations, including the lining, with a fabric pen as you remove them.
Clip the threads along the skirt seams, where the fabric and lining are sewn together, and pull the pieces apart.
Spread out the new upholstery fabric wrong side up. Arrange the original upholstery pieces on top of the fabric, wrong side up. Trace around the original pieces with a fabric pen. Repeat with the muslin for the lining pieces.
Cut out the upholstery pieces along the traced lines. Label them with a fabric pen to match the original markings. Draw the seam, fold and staple marks from the original pieces on the new fabric as well. Repeat with the muslin for the lining pieces.
Spread out the skirt upholstery piece right side up. Align the matching muslin lining piece on top of the upholstery fabric. Pin them together along the bottom seam marking.
Sew the skirt pieces together along the pinned edge and remove the pins. Fold the fabric in half along the seam, turning it right side out.
Place the top upholstery piece on top of the hassock, right side up. Staple one edge of the fabric to the side of the hassock approximately 1/4 inch below the original piping location.
Rotate the hassock 180 degrees. Pull the fabric taut across the top and staple down the edge. Turn the hassock and staple down the edges of the fabric until it is secure around the hassock.
Place one end of the piping against the side of the hassock with the rounded edge up. Align the piping's seam 1/4 inch higher than the stapled edge of the top upholstery fabric. Staple the piping through the seam and into the hassock. Staple the remainder of the piping around the hassock. Turn the end down and staple it. Cut off the excess piping.
Turn the hassock upside down. Hold the left edge of the skirt against the side of the hassock with the lining out and the raw edge up.
Fold the left edge of the skirt over toward you, top to bottom, approximately 1/2 inch. Align the original staple or tack markings on the skirt with the stapled edge of the piping. Staple through the folded edge of the skirt and into the stool at the mark. Staple the remainder of the skirt around the stool until you are within 2 inches of the starting point.
Fold the right edge of the skirt toward you, top to bottom, and adjust the fold wider or narrower until it meets or slightly overlaps the fold of the right edge of the skirt. Staple through the fold and into the stool.
Turn the stool over. Stitch the two open edges of the skirt together by hand at the hem.