The presence of charge and spin stripe order in the La2CuO4-based family of superconductors continues to lead to new insight on the unusual ground state properties of high Tc cuprates. Soon after the discovery of charge stripe order at T(charge)=65K in Nd3+ co-doped LSCO ($T_{c}simeq6$~K) [Tranquada et al., Nature {bf 375} (1995) 561], Hunt et al. demonstrated that La$_{1.48}$Nd$_{0.4}$Sr$_{0.12}$CuO$_4$ and superconducting LSCO with x~1/8 (Tc ~ 30K) share nearly identical NMR anomalies near $T_{charge}$ of the former [Phys. Rev. Lett. {bf 82} (1999) 4300]. Their inevitable conclusion that LSCO also undergoes charge order at a comparable temperature became controversial, because diffraction measurements at the time were unable to detect Bragg peaks associated with charge order. Recent advances in x-ray diffraction techniques finally led to definitive confirmations of the charge order Bragg peaks in LSCO with an onset at as high as T(charge)=80K. Meanwhile, improved instrumental technology has enabled routine NMR measurements that were not feasible two decades ago. Motivated by these new developments, we revisit the charge order transition of a LSCO single crystal based on 63Cu NMR techniques. We demonstrate that 63Cu NMR properties of the nuclear spin $I_{z}$ = -1/2 to +1/2 central transition below T(charge) exhibit unprecedentedly strong dependence on the measurement time scale set by the NMR pulse separation time $tau$; a new kind of anomalous, very broad wing-like 63Cu NMR signals gradually emerge below T(charge) only for extremely short $tau lesssim 4~mu$s, while the spectral weight of the normal NMR signals is progressively wiped out. The NMR linewidth and relaxation rates depend strongly on $tau$ below T(charge), and their enhancement in the charge ordered state indicates that charge order turns on strong but inhomogeneous growth of Cu spin-spin correlations.