In this Letter, we report the first observation of dramatic decrease in nanosecond (ns) pulsed laser-induced transient absorption (TA) in a-Ge(x)As(35-x)Se₆₅ thin films by tuning the amorphous network from floppy to rigid. Our results provide the direct experimental evidence of a self-trapped exciton mechanism, where trapping of the excitons occurs through bond rearrangements. Taken together, a rigid amorphous network with more constraints than degrees of freedom are unable to undergo any such bond rearrangements and results in weaker TA. However, we also demonstrate that excitation fluence can be effectively utilized as a simple tool to lift up enough constraints to introduce large TA even in rigid networks. Apart from this, we also show that TA is tunable with network rigidity as it blueshifts when the mean coordination is increased from 2.35 to 2.6.