There are multiple stall speeds that have been defined. For example, the 14 CFR §1.2 Abbreviations and symbols lists the following stall speeds:
$V_{S}$ means the stalling speed or the minimum steady flight speed at which the airplane is controllable.
$V_{SO}$ means the stalling speed or the minimum steady flight speed in the landing configuration.
$V_{S1}$ means the stalling speed or the minimum steady flight speed obtained in a specific configuration.
$V_{SR}$ means reference stall speed.
$V_{SRO}$ means reference stall speed in the landing configuration.
$V_{SR1}$ means reference stall speed in a specific configuration.
$V_{SW}$ means speed at which onset of natural or artificial stall warning occurs.
What you're asking is the $V_{S}$, the minimum flight speed at which the airplane is controllable. In short, as $V_{stall} \propto \sqrt{\frac{1}{C_{L}}}$, $C_{L_{max}}$ gives the minimum speed at which the aicraft is controllable.
Consider an aircraft in a level flight, If the pilot wants to reduce the speed, in order to maintain a steady, level flight, he/she has to increase the angle of attack i.e. increase the $C_{L}$. He can do this till the $C_{L}$ reaches $C_{L_{max}}$, where the speed becomes minimum while the aircraft is still in a steady, level flight. If the speed is reduced any further, the aircraft loses lift; this speed gives the stall speed of the aircraft.
You can get the stall speed for any configuration by using the $C_{L}$ at that configuration; but the values have to be realistic. Setting the $C_{L}$ to very low values to get large $V_{stall}$ makes no physical or practical sense. For example, the aircraft can be set to have zero (or even negative) $C_{L_{min}}$, in which case, the stall speed has no meaning.