I ended-up having to take a information from several different places to create a solution that works for me (in reality, the beginnings of a production viable solution - but it works!)
I got rid of JwtAuthForWebAPI (though I did borrow one piece from it to allow requests with no Authorization header to flow through to WebAPI Controller methods not guarded by [Authorize]).
Instead I'm using Microsoft's JWT Library (JSON Web Token Handler for the Microsoft .NET Framework - from NuGet).
In my authentication method, after doing the actual authentication, I create the string version of the token and pass it back along with the authenticated name (the same username passed into me, in this case) and a role which, in reality, would likely be derived during authentication.
Here's the method:
[HttpPost]
public LoginResult PostSignIn([FromBody] Credentials credentials)
{
    var auth = new LoginResult() { Authenticated = false };
    if (TryLogon(credentials.UserName, credentials.Password))
    {
        var tokenDescriptor = new SecurityTokenDescriptor
        {
            Subject = new ClaimsIdentity(new[]
            {
                new Claim(ClaimTypes.Name, credentials.UserName), 
                new Claim(ClaimTypes.Role, "Admin")
            }),
            AppliesToAddress = ConfigurationManager.AppSettings["JwtAllowedAudience"],
            TokenIssuerName = ConfigurationManager.AppSettings["JwtValidIssuer"],
            SigningCredentials = new SigningCredentials(new 
                InMemorySymmetricSecurityKey(JwtTokenValidationHandler.SymmetricKey),
                "http://www.w3.org/2001/04/xmldsig-more#hmac-sha256",
                "http://www.w3.org/2001/04/xmlenc#sha256")
            };
            var tokenHandler = new JwtSecurityTokenHandler();
            var token = tokenHandler.CreateToken(tokenDescriptor);
            var tokenString = tokenHandler.WriteToken(token);
            auth.Token = tokenString;
            auth.Authenticated = true;
        }
    return auth;
}
UPDATE
There was a question about handling the token on subsequent requests.  What I did was create a DelegatingHandler to try and read/decode the token, then create a Principal and set it into Thread.CurrentPrincipal and HttpContext.Current.User (you need to set it into both).  Finally, I decorate the controller methods with the appropriate access restrictions.
Here's the meat of the DelegatingHandler:
private static bool TryRetrieveToken(HttpRequestMessage request, out string token)
{
    token = null;
    IEnumerable<string> authzHeaders;
    if (!request.Headers.TryGetValues("Authorization", out authzHeaders) || authzHeaders.Count() > 1)
    {
        return false;
    }
    var bearerToken = authzHeaders.ElementAt(0);
    token = bearerToken.StartsWith("Bearer ") ? bearerToken.Substring(7) : bearerToken;
    return true;
}
protected override Task<HttpResponseMessage> SendAsync(HttpRequestMessage request, CancellationToken cancellationToken)
{
    HttpStatusCode statusCode;
    string token;
    var authHeader = request.Headers.Authorization;
    if (authHeader == null)
    {
        // missing authorization header
        return base.SendAsync(request, cancellationToken);
    }
    if (!TryRetrieveToken(request, out token))
    {
        statusCode = HttpStatusCode.Unauthorized;
        return Task<HttpResponseMessage>.Factory.StartNew(() => new HttpResponseMessage(statusCode));
    }
    try
    {
        JwtSecurityTokenHandler tokenHandler = new JwtSecurityTokenHandler();
        TokenValidationParameters validationParameters =
            new TokenValidationParameters()
            {
                AllowedAudience = ConfigurationManager.AppSettings["JwtAllowedAudience"],
                ValidIssuer = ConfigurationManager.AppSettings["JwtValidIssuer"],
                SigningToken = new BinarySecretSecurityToken(SymmetricKey)
            };
        IPrincipal principal = tokenHandler.ValidateToken(token, validationParameters);
        Thread.CurrentPrincipal = principal;
        HttpContext.Current.User = principal;
        return base.SendAsync(request, cancellationToken);
    }
    catch (SecurityTokenValidationException e)
    {
        statusCode = HttpStatusCode.Unauthorized;
    }
    catch (Exception)
    {
        statusCode = HttpStatusCode.InternalServerError;
    }
    return Task<HttpResponseMessage>.Factory.StartNew(() => new HttpResponseMessage(statusCode));
}
Don't forget to add it into the MessageHandlers pipeline:
public static void Start()
{
    GlobalConfiguration.Configuration.MessageHandlers.Add(new JwtTokenValidationHandler());
}
Finally, decorate your controller methods:
[Authorize(Roles = "OneRoleHere")]
[GET("/api/admin/settings/product/allorgs")]
[HttpGet]
public List<Org> GetAllOrganizations()
{
    return QueryableDependencies.GetMergedOrganizations().ToList();
}
[Authorize(Roles = "ADifferentRoleHere")]
[GET("/api/admin/settings/product/allorgswithapproval")]
[HttpGet]
public List<ApprovableOrg> GetAllOrganizationsWithApproval()
{
    return QueryableDependencies.GetMergedOrganizationsWithApproval().ToList();
}