ABSTRACT The angular momentum (AM) content of massive stellar cores helps us to determine the natal spin rates of neutron stars and black holes. Asteroseismic measurements of low-mass stars have proven that stellar cores rotate slower than predicted by most prior work, so revised models are necessary. In this work, we apply an updated AM transport model based on the Tayler instability to massive helium stars in close binaries, in which tidal spin-up can greatly increase the star’s AM. Consistent with prior work, these stars can produce highly spinning black holes upon core-collapse if the orbital period is less than $P_{\rm orb} \lesssim \! 1 \, {\rm d}$. For neutron stars, we predict a strong correlation between the pre-explosion mass and the neutron star rotation rate, with millisecond periods ($P_{\rm NS} \lesssim 5 \, {\rm ms}$) only achievable for massive ($M \gtrsim 10 \, M_\odot$) helium stars in tight ($P_{\rm orb} \lesssim 1 \, {\rm d}$) binaries. Finally, we discuss our models in relation to type Ib/c supernovae, superluminous supernove, gamma-ray bursts, and LIGO/Virgo measurements of black hole spins. Our models are roughly consistent with the rates and energetics of these phenomena, with the exception of broad-lined Ic supernovae, whose high rates and ejecta energies are difficult to explain.