As the maintainer of the Kerberos.NET as mentioned in the other answer, you can always go that route. However, if you want SSO using the currently signed in Windows identity you have to go through Windows' SSPI stack.
Many .NET clients already support this natively using Windows Integrated auth, its just a matter of finding the correct knobs. It's unclear what client you're using so I can't offer any suggestions beyond that.
However if this is a custom client you have to call into SSPI directly. There's a handful of really good answers for explaining how to do that such as: Client-server authentication - using SSPI?.
The aforementioned Kerberos.NET library does have a small wrapper around SSPI: https://github.com/dotnet/Kerberos.NET/blob/develop/Kerberos.NET/Win32/SspiContext.cs
It's pretty trivial:
using (var context = new SspiContext($"host/yourremoteserver.com", "Negotiate"))
{
var tokenBytes = context.RequestToken();
var header = "Negotiate " + Convert.ToBase64String(tokenBytes);
...
}