  We investigate the low-temperature properties of the Ising-like screw chain
antiferromagnet SrCo$_2$V$_2$O$_8$ under a longitudinal magnetic field by
susceptibility and $^{51}$V NMR measurements. The bulk susceptibility $\chi$
shows an onset of long-range Ising-antiferromagnetic (AFM) order and the
suppression of the order by field with the N\'{e}el temperature dropped from
5.1~K to 2~K when field increases from 0.1~T to 4~T. The suppression of the AFM
order by the field is also observed by the NMR spectra and the spin-lattice
relaxation $1/T_1$. At fields above 4~T, $\chi$ shows a low-temperature upturn,
which is consistent with the onset of a transverse antiferromagnetic order as
supported by the quantum Monte Carlo simulations. A line splitting in the NMR
spectra is also observed at high temperatures. We show that the line split
characterizes the onset of a short-range transverse antiferromagnetic order
with magnetic moments orientated along the crystalline [110]/[1$\bar1$0]
directions. The $1/T_1$ data at higher temperature show a power-law behavior
$1/T_1{\sim}T^{\alpha}$, which is consistent with the Tomonaga-Luttinger-liquid
behavior. With increasing the field, the power-law exponent $\alpha$ changes
from negative to positive, which clearly shows an inversion of the Luttinger
exponent $\eta$, where the dominant low-energy spin fluctuations switch from
the longitudinal type to the transverse type at a high field of 7~T.
