  We introduce a method to compute one-loop soft functions for exclusive
$N$-jet processes at hadron colliders, allowing for different definitions of
the algorithm that determines the jet regions and of the measurements in those
regions. In particular, we generalize the $N$-jettiness hemisphere
decomposition of [Jouttenus 2011] in a manner that separates the dependence on
the jet boundary from the observables measured inside the jet and beam regions.
Results are given for several factorizable jet definitions, including
anti-$k_T$, XCone, and other geometric partitionings. We calculate explicitly
the soft functions for angularity measurements, including jet mass and jet
broadening, in $pp \to L + 1$ jet and explore the differences for various jet
vetoes and algorithms. This includes a consistent treatment of rapidity
divergences when applicable. We also compute analytic results for these soft
functions in an expansion for a small jet radius $R$. We find that the
small-$R$ results, including corrections up to $\mathcal{O}(R^2)$, accurately
capture the full behavior over a large range of $R$.
