Using a sample of spiral galaxies selected from the Sloan Digital Sky Survey Data Release 7 and Galaxy Zoo 2, we investigate the alignment of spin axes of spiral galaxies with their surrounding large-scale structure, which is characterized by the large-scale tidal field reconstructed from the data using galaxy groups above a certain mass threshold. We find that the spin axes only have weak tendencies to be aligned with (or perpendicular to) the intermediate (or minor) axis of the local tidal tensor. The signal is the strongest in a cluster environment where all three eigenvalues of the local tidal tensor are positive. Compared to the alignments between halo spins and the local tidal field obtained in N-body simulations, the above observational results are in best agreement with those for the spins of inner regions of halos, suggesting that the disk material traces the angular momentum of dark matter halos in the inner regions.