in the region R < r < 2R, the magnetic field is along the common axis
in the region R < r < 2R, the magnetic field is tangential to the circle of radius r, centered on the axis
in the region 0 < r < R, the magnetic field is non-zero
in the region r > 2R, the magnetic field is non-zero
From r:0-R the magnetic field will be of solenoid since magnetic field due to the current carrying wire is zero from amperes law as there is no current in region. Due to the field of solenoid, the overall magnetic field is non zero in region 0 < r < R along it's axis.