  Theories which provide a dynamical explanation for the large top-quark mass
often include TeV-scale vector-like top-quark and bottom-quark partner states
which can be potentially discovered at the LHC. These states are currently
probed through model-independent searches for pair-production via gluon fusion,
as well as through model-dependent complementary electroweak single production.
In this paper we study the potential to extend those searches for the partners
of the third-generation Standard Model quarks on the basis of their expected
chromomagnetic interactions. We discuss how current searches for "excited"
bottom-quarks produced via $b$-gluon fusion through chromomagnetic interactions
are relevant, and provide significant constraints. We then explore the region
of the parameter space in which the bottom-quark partner is heavier than the
top-quark partner, in which case the top-partner can be primarily produced via
the decay of the bottom-partner. Next, we probe the potential of the production
of a single top-quark partner in association with an ordinary top-quark by
gluon-fusion. Kinematically these two new processes are similar, and they yield
the production of a heavy top partner and a lighter Standard Model state, a
pattern which allows for the rejection of the associated dominant Standard
Model backgrounds. We examine the sensitivity of these modes in the case where
the top-partner subsequently decays to a Higgs boson and an ordinary top-quark,
and we demonstrate that these new channels have the potential of extending and
complementing the conventional strategies at LHC run III and at the
high-luminosity phase of the LHC. In this last case, we find that partner
masses that range up to about 3 TeV can be reached. This substantially expands
the expected mass reach for these new states, including regions of parameter
space that are inaccessible by traditional searches.
