  The Hamiltonian analysis for $f(T)$ gravity implies the existence of at least
one scalar-type degree of freedom (DoF). However, this scalar DoF of $f(T)$
gravity does not manifest in linear perturbations around a cosmological
background, which indicates an underlying strong coupling problem. In this work
we expand the scope by introducing an extra scalar field non-minimally coupled
to $f(T)$ gravity, aiming to address or alleviate the aforementioned strong
coupling problem. Employing the effective field theory (EFT) approach, we
provide a class of torsional EFT forms up to second order operators, avoiding
the Ostrogradsky ghost. To illustrate this phenomenon, we study a simple model
and perform a detailed analysis of its linear scalar perturbations. The results
demonstrate that the coupling terms in this toy model are necessary to avoid
the initial degenerate situation. The complete avoidance of new constraints
requires more coupling terms. Once this vanishing scalar DoF starts propagating
in cosmological background at linear level, this phenomenon will demand a
revisit of the strong coupling issue that arises in $f(T)$ gravity,
particularly in the presence of matter coupling.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics