Wildcard SSL Certificate can secure only first level sub-domains of the common name (CN), so when the certificate issued for *.example-private.com it can secure the root domain and it's all first level sub-domains as below.
To secure second level sub-domains as *.staging.example-private.com, you have two options.
1. Purchase another wildcard certificate:
Your administrator needs to purchase the different certificates for each one and manage all certificates configuration, renewal, expiry, and installation.
2. Purchase Multi-Domain wildcard certificate:
This product allows you to secure up to 100 websites and its unlimited sub-domains with a single certificate. You need to set *.example-private.com as your primary domain name and add another domain names in the SAN field.
For more information about how multi-domain wildcard certificate works, you can refer this link - https://www.ssl2buy.com/comodo-multi-domain-wildcard-ssl.php