An apparatus for winding a coil pack of a fiber such as an optical fiber winds the fiber as a radial coil, and then joins the coil axially to previously wound coils. Each radial coil is wound on a flexible flange that is deformable from a first tapered shape which facilitates the winding against the flange, to a second shape wherein the coil wound thereon conforms to the shape of the last coil affixed to the fiber pack. The flexible flange is preferably made of an elastomer, and its shape change is readily accomplished by an articulated series of nesting cylinders. The apparatus further includes a fiber winding head that winds the fiber onto the flexible flange, and an adhesive applicator head that applies adhesive between succeeding coils of fiber as they are joined to the coil pack.