  As usual, we observe an unknown coupling function, i.e. $F(\varphi)$, with a
function of torsion and also curvature, i.e. $f(T)$ and $f(R)$, generally
depending on a scalar field. In $f(R)$ case, it comes from quantum correlations
and other reasons. Now, what if beside this term in $f(T)$-gravity context, we
enhance the action through another term which depends on both scalar field and
its derivatives? In this paper, we have added such an unprecedented term in the
generic common action of $f(T)$-gravity such that in this new term, an unknown
function of torsion has coupled with an unknown function of both scalar field
and its derivatives. We have explained why we can append such term, in the
introduction in details. By the use of Noether symmetry approach, we have
considered its behavior and effect. We have shown that it does not produce an
anomaly, but rather it works successfully and numerical analysis of the exact
solutions of field equations coincide with all most important observational
data particularly late-time-accelerated expansion. So, this new term may be
added to the gravitational actions of $f(T)$-gravity.
