Understanding the magnetic field structure of spicules is essential for developing their formation models, which are important for understanding the mass transport to the corona and coronal heating. We report the spatial distribution of spicule magnetic fields from spectropolarimetric observations with SUNRISE III/SCIP, which provides high spatial resolution and high polarization sensitivity under seeing-free conditions. We observed the solar limb above a quiet region with SCIP and detected clear Stokes V signals in the Ca II 8542 Å line from off-limb spicules. We applied the weak field approximation to estimate the line-of-sight (LOS) component of the magnetic field (BLOS) of spicules. We find that BLOS ranges up to 20 G, nearly independent of height, in the region up to 3″ from the limb, whereas the distribution broadens to 40 G at 3″─6″ from the limb. The Stokes I profiles averaged at increasing distance from the limb reveal a transition from a double-peaked shape to a single-peaked one at around 3″ from the limb. Because we can see a dense forest of spicules overlapping each other above the limb, the observational results suggest that the BLOS depends on how deeply the Ca II line can penetrate the spicule forest along the LOS. At heights with double-peaked profiles, the estimated BLOS represents the magnetic field of a foreground spicule. In contrast, at heights with single-peaked profiles, it reflects contributions from a number of prominent, tall, inclined spicules, leading to more dispersed BLOS values.