  The rapidity-dependent jet veto observables $\mathcal{T}_{Bj}$ and
$\mathcal{T}_{Cj}$ provide a tight jet veto at central rapidity, gradually
transitioning to a loose veto at forward rapidities. They divide the phase
space into exclusive jet bins in a different way to the traditional jet veto
observable $p_{Tj}$, and are advantageous to use under harsh pile-up
conditions. We obtain predictions for the $0$-jet gluon-fusion (ggF) Higgs
cross section using both of these veto observables at NNLL$'+$NNLO, and compare
these predictions to the prior state-of-the-art of NLL$'+$NLO. A significant
reduction in perturbative uncertainty is observed going from NLL$'+$NLO to
NNLL$'+$NNLO, with the NNLL$'+$NNLO predictions lying inside the uncertainty
band of the NLL$'+$NLO predictions. We also investigate the relative
sensitivities of ggF Higgs cross sections with $\mathcal{T}_{Bj}$,
$\mathcal{T}_{Cj}$ and $p_{Tj}$ jet vetoes to underlying event and
hadronisation effects using an NLO+parton shower calculation. We find that the
cross sections with $\mathcal{T}_{Bj}$ and $\mathcal{T}_{Cj}$ vetoes have a
reduced sensitivity to underlying event and hadronisation effects compared to
that with a $p_{Tj}$ veto.
