Can you restrict the services a certificate can access on the client side?

I have a verification certificate signed by my organisation’s CA, which I can use to authenticate my user account on intranet web services.

Is there someway I can sign a new certificate which can only authenticate to one specific web service. Or some other way to enable limited access to one web service by a script I don’t want to give full access to my verification certificate.

Unfortunately I don’t have access to modify the web service, which is running nginx.