In the temperature-magnetic field phase diagram, the binary metallic compound MnSi exhibits three magnetic phases below T_c ≈ 29 K. An unconventional helicoidal phase is observed in zero field. At moderate field intensity a conical phase sets in. Near T_c, in an intermediate field range, a skyrmion lattice phase appears. Here we show the magnetic structure in the conical phase to strongly depend on the field direction and to deviate substantially from a conventional conical structure.