configure Unity to resolve a constructor parameter and interface

I am looking for a solution where I don't have to change the FaxService constructor parameters.

There is nothing built into Unity to do this without specifying at least the types of all constructor parameters.

var container = new UnityContainer();
var phone = "214-123-4567";
container.RegisterType<IFaxService, FaxService>(new InjectionConstructor(phone, typeof(IFaxProvider)));
container.RegisterType<IFaxProvider, EFaxProvider>();

var fax = container.Resolve<IFaxService>();

The list of parameters would need to change every time the constructor changes.

Workaround

If you don't want to change your DI registration if the constructor changes for FaxService, build an abstract factory to separate the string/primitive parameters from the services. Pass services through the constructor parameters of the factory. Pass string and primitive parameter types through the Build method parameters.

public class FaxServiceFactory
{
    private readonly IFaxProvider faxProvider;
    public FaxServiceFactory(IFaxProvider faxProvider)
    {
        this.faxProvider = faxProvider ?? throw new ArgumentNullException(nameof(faxProvider));
    }

    public IFaxService Create(string phone)
    {
        return new FaxService(phone, this.faxProvider);
    }
}

And then register like:

var container = new UnityContainer();

var phone = "214-123-4567";
container.RegisterType<FaxServiceFactory>();
container.RegisterType<IFaxProvider, EFaxProvider>();
container.RegisterInstance<IFaxService>(container.Resolve<FaxServiceFactory>().Create(phone));

var fax = container.Resolve<IFaxService>();

Another possibility is to use a DI container that allows you to specify partial lists of constructor parameters (Autofac, Ninject, StructureMap, and Castle Windsor all do this out of the box).