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.
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