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Restructuring and Integration Expense - Schedule of Severance and Employee Benefit Costs (Details) - USD ($)
$ in Millions
 3 Months Ended 9 Months Ended
Mar. 02, 2024
Mar. 04, 2023
Mar. 02, 2024
Mar. 04, 2023
Changes in Restructuring Liability Balance [Roll Forward]        
Integration Costs $ 1.7 $ 4.6 $ 8.7 $ 19.8
Knoll Integration        
Changes in Restructuring Liability Balance [Roll Forward]        
Integration Costs 7.6 4.0 18.4 12.7
Total | Knoll Integration        
Changes in Restructuring Liability Balance [Roll Forward]        
Beginning balance     2.9  
Integration Costs     15.3  
Amounts Paid     (13.0)  
Non-cash costs     (3.9)  
Ending balance 1.3   1.3  
Severance and Employee Benefit | Knoll Integration        
Changes in Restructuring Liability Balance [Roll Forward]        
Beginning balance     2.9  
Integration Costs   1.7 0.0 3.1
Amounts Paid     (2.8)  
Non-cash costs     0.0  
Ending balance 0.1   0.1  
Exit and Disposal Activities | Knoll Integration        
Changes in Restructuring Liability Balance [Roll Forward]        
Beginning balance     0.0  
Integration Costs 6.6 $ 0.8 15.3 $ 3.6
Amounts Paid     (10.2)  
Non-cash costs     (3.9)  
Ending balance $ 1.2   $ 1.2