A method for manufacturing a tire includes making at least one semi-finished elementary product of elastomeric material and depositing it in adjacent turns on a toroidal support. Making the at least one elementary product includes extruding the elastomeric material through an extruder head including a bushing defining a tapered extrusion channel. An extruder includes a chamber, at least one helical screw disposed in the chamber, and the extruder head. The head includes an outer tubular body attached by one end to the chamber, a bushing disposed inside the body to define the tapered extrusion channel, and a die disposed inside the body. A ratio of an area of the inlet section to an area of the outlet section is greater than 5:1 and less than 100:1 and a ratio of a diameter of the inlet section to a length of the bushing is greater than 0.1 and less than 1:1.