  Recently more generalized four-fermion interactions of neutrinos such as
tensor and scalar interactions (TSIs) have been extensively studied in response
to forthcoming precision measurements of neutrino interactions. In this letter,
we show that due to the chirality-flipping nature, at the 1-loop level TSIs
typically generate much larger ($10^{7}\sim10^{10}$) neutrino magnetic moments
($\nu$MMs) than the vector case. For some cases, the large $\nu$MMs generated
by TSIs may reach or exceed the known bounds, which implies potentially
important interplay between probing TSIs and searching for $\nu$MMs in current
and future neutrino experiments.
