Consider how a loudspeaker is made. You have an external cylindrical
magnet with poles at either end. One end A has a flat iron plate with a large
hole in it. The other end B is fitted with another flat plate with a central
solid iron rod which projects axially to the top of the hole in the plate A. The
coil moves in the radial annular field between plate A and the central iron rod.
Can you get a mild steel tube which will surround the coil, a flat 1/4" mild steel plate and a stub of 3/4" mild steel rod with flat machined ends? Lap the end of the tube flat with emery paper on the plate. Sit the magnets on top of the rod and the rod on the centre of the plate. Put the iron tube centrally around the magnets and lower in the coil so that the top face of the magnets is about half way down it.
Alternatively, get three strips of 1/16" mild steel, bend them in a U
shape and mount the mild steel column in the centre, with the strips at 120
degrees. Cut the strips to length so that when stacked they end level with the
top of the magnets. You could also use a taller stack of magnets, but 2 off 1/4"
ones won't reach the centre of the coil and allow for any free motion. You will
probably need to stick the strips together with acrylic glue to form a rigid cup
structure. Then give it a coat of anti rust paint.
For an alternative design, see the sensor coil on the Lehman page.