lightning:omniChannelLogout can be listened in the Salesforce Lighting Aura Component to listen to the Salesforce Omni-Channel logouts.

Sample Lightning Aura Component:

Component:

<aura:component 
                implements="flexipage:availableForAllPageTypes" 
                access="global" >
    <lightning:omniToolkitAPI 
                              aura:id="omniToolkit" />  
    <aura:handler 
                  event="lightning:omniChannelLogout" 
                  action="{! c.onLogout }"/>
</aura:component>

Controller:

( {
    
    onLogout : function( component, event, helper ) {
        
        let strReason = event.getParam(
            'reason'
        );
        console.log(
        	'The reason for Omni-Channel logout is',
            strReason
        );
        
    }
    
} )

Add the Lighting Component to the App to listen to the Salesforce Omni-Channel logouts.

Output: