  The quantum criticality of an Ising-like screw chain antiferromagnet
SrCo$_2$V$_2$O$_8$, with a transverse magnetic field applied along the
crystalline $a$-axis, is investigated by ultra-low temperature NMR
measurements. The N\'{e}el temperature is rapidly and continuously suppressed
by the field, giving rise to a quantum critical point (QCP) at
$H_{C{_1}}$$\approx$~7.0~T. Surprisingly, a second QCP at
$H_{C{_2}}\approx$~7.7~T featured with gapless excitations is resolved from
both the double-peak structure of the field dependent spin-lattice relaxation
rate $1/^{51}T_1$ at low temperatures and the weakly temperature-dependent
$1/^{51}T_1$ at this field. Our data, combined with numerical calculations,
suggest that the induced effective staggered transverse field significantly
lowers the critical fields, and leads to an exposed QCP at $H_{C{_2}}$, which
belongs to the one-dimensional transverse-field Ising universality.
