Goodwill and Intangible Assets - Narrative (Details) - USD ($)		12 Months Ended
	Jul. 04, 2022	Oct. 02, 2022	Oct. 03, 2021	Sep. 27, 2020
Goodwill [Line Items]
Impairment of goodwill	$ 0	$ 0	$ 0	$ 15,800,000
Percentage of excess of fair value over carrying value (less than)	165.00%
Amortization expense for intangible assets		13,200,000	11,500,000	$ 11,600,000
Reduction in intangible assets, foreign currency translation adjustment		5,300,000
GSG
Goodwill [Line Items]
Gross amounts of goodwill		536,800,000	556,100,000
Accumulated impairment		17,700,000	17,700,000
CIG
Goodwill [Line Items]
Gross amounts of goodwill		712,800,000	691,600,000
Accumulated impairment		$ 121,500,000	$ 121,500,000