Thought I'd give an update, I was able to get this code working and all is good now.  I took my SOQL out of the constructor and moved it down to the SetEmployeeName() method.  Then I captured the incoming URL via ApexPages.currentPage().getParameters() and then parsed out the fields for the Master object.  Then it was a matter of appending those to the new URL and redirecting.  Full code below, hope someone finds this helpful!

Visualforce page:

<apex:page standardController="Lab_Project_Hours__c" extensions="ControllerSetLabProjectHourEmployeeName" action="{!SetLabProjectHourEmployeeName}" >

</apex:page>

Apex Class:

public class ControllerSetLabProjectHourEmployeeName{

  private final Lab_Project_Hours__c objLabProject;

  // Constructor method
  public ControllerSetLabProjectHourEmployeeName(ApexPages.StandardController stdController){    
     this.objLabProject = (Lab_Project_Hours__c) stdController.getRecord();
  }
  
  public string EmployeeName{ get; set; } 

  public string EmployeeId{ get; set; } 
  
  // This method is used to set employee name and redirect to project hours new page
  public PageReference SetLabProjectHourEmployeeName(){
    this.EmployeeName = '';
    this.EmployeeId = '';
    
    Map<String, String> parentURL = ApexPages.currentPage().getParameters();
    
    String ProjectId = parentURL.get('CF00N0000000715oD_lkid');   
    String ProjectRecordTypeId = parentURL.get('RecordType'); 
    String ProjectName = parentURL.get('CF00N0000000715oD');
             
     try{
      User currentUser = [Select Id, Employee_Page__c From User Where Id = : UserInfo.getUserId() Limit 1];
      if(currentUser.Employee_Page__c != null){
         List<Employee__c> listEmployee = [Select Id, Name From Employee__c where Id = : currentUser.Employee_Page__c Limit 1];
         if(listEmployee.size() > 0){
           this.EmployeeId = listEmployee.get(0).Id;
           this.EmployeeName = listEmployee.get(0).Name;
         }  
      } 
    }
    catch(Exception ex){
      system.debug(ex.getMessage());
    }    
    
    PageReference pageRef;
    
    if(ProjectRecordTypeId != null){
        pageRef = new PageReference('/a0G/e?nooverride=1&retURL=/' + ProjectId + '&CF00N00000008W0jP_lkid=' + this.EmployeeId + '&CF00N00000008W0jP=' + this.EmployeeName + '&CF00N0000000715oD_lkid=' + ProjectId + '&CF00N0000000715oD=' + ProjectName + '&RecordType=' + ProjectRecordTypeId);
    } 
    else {
        pageRef = new PageReference('/a0G/e?nooverride=1&retURL=/' + ProjectId + '&CF00N00000008W0jP_lkid=' + this.EmployeeId + '&CF00N00000008W0jP=' + this.EmployeeName + '&CF00N0000000715oD_lkid=' + ProjectId + '&CF00N0000000715oD=' + ProjectName);
    }

    pageRef.setRedirect(true);
    return pageRef;
  }
}