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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 |

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- Definition Amount of expense for award under share-based payment arrangement. Excludes amount capitalized. + References Reference 1: http://www.xbrl.org/2003/role/disclosureRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (h)(1)(i) -URI https://asc.fasb.org/extlink&oid=128089324&loc=d3e5070-113901 + Details
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