We report the results of a muon-spin relaxation ($mu$SR) investigation of La$_{2-x}$Sr$_{x}$CoO$_{4}$, an antiferromagnetic insulating series which has been shown to support charge ordered and magnetic stripe phases and an hourglass magnetic excitation spectrum. We present a revised magnetic phase diagram, which shows that the suppression of the magnetic ordering temperature is highly sensitive to small concentrations of holes. Distinct behavior within an intermediate $x$ range ($0.2 leq x lesssim 0.6$) suggests that the putative stripe ordered phase extends to lower $x$ than previously thought. Further charge doping ($0.67 leq x leq 0.9$) prevents magnetic ordering for $T gtrsim 1.5~{rm K}$
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