We report on the existence and stability of hairy wormlike micelles, which are obtained by adding in a surfactant solution, an amphiphilic copolymer, whose hydrophobic block anchors onto the micelles and whose hydrophilic tails are swollen in the aqueous solvent. Small angle neutron scattering is performed to characterize the structural properties of the copolymer/surfactant mixture. We first show that the locally cylindrical structure of the micelles is maintained upon copolymer addition over a wide range of copolymer-to-surfactant molar ratio. On the other hand, the addition of amphiphilic copolymer correlates with the emergence of a broad peak in the structure factor at a finite wave vector. The peak can be related to a liquidlike order of the micelles due to steric repulsions induced by the copolymer layer covering the micelles. Finally, we show that the phase diagram of the hairy micelles can be quantitatively interpreted in terms of an effective volume fraction of the cylinders, assuming that the radius of the cylinders is increased by the thickness of the surrounding polymer layer. We find that this thickness is of the same order of magnitude as the radius of gyration of the decorating copolymer and increases with the polymeric layer density.