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Stock-based Compensation and Equity Grants - Stock-based Compensation Expense (Details) - USD ($)
$ in Thousands
3 Months Ended 12 Months Ended
Dec. 31, 2021
Sep. 30, 2022
Sep. 30, 2021
Selling, General and Administrative Expenses      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Total compensation expense   $ 2,608 $ 1,660
DLH Employees      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Unrecognized expense for DLH employees (a)   $ 5,214 4,468
Weighted average period for recognition of compensation expense   4 years 2 months 12 days  
DLH Employees | Selling, General and Administrative Expenses      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Total compensation expense   $ 1,960 1,193
Non-employee Directors | Restricted Stock Units      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Number of restricted stock units issued (in shares) 53,510    
Number of restricted stocks vested in period (in shares)   53,510  
Non-employee Directors | Selling, General and Administrative Expenses      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Total compensation expense   $ 648 $ 467
NEO | Restricted Stock Units      
Employee Service Share-based Compensation, Allocation of Recognized Period Costs [Line Items]      
Number of restricted stock units issued (in shares)   140,404  
Number of restricted stock outstanding (in shares)   140,404