TRESTClient capture a reasonably exception cause, e.g. when server is not available

Question

I'm struggeling with how to capture the exception cause when I send a RESTRequest to a server which isn't available at the time of the request.

I do get an exception, but it's really a poor exception to work with when I need to present a error message dialog to the user and e.g. disconnect the client.

Question is: How am I supposed to capture the "right" exception in my application, the details of which are "hidden" and end up as a relatively useless ERESTException?

This is not an EHTTPProtocolException, so the event on the client or request will not occur on this error.

BTW. Using Delphi 10.2.1.

The ERESTException.Message I receive back, looks like this:

Exception REST request failed: Error sending data: (12029) A connection with the server could not be established

I'm using TRESTClient, TRESTRequest and TRESTResponse.

When I execute a request to an offline REST service it fails (of course)

System.Net.HttpClient.Win

function TWinHTTPClient.DoExecuteRequest(const ARequest: THTTPRequest; var AResponse: THTTPResponse;
  const AContentStream: TStream): TWinHTTPClient.TExecutionResult;
.....

  // Send Request
  LRes := WinHttpSendRequest(LRequest.FWRequest, WINHTTP_NO_ADDITIONAL_HEADERS, 0, WINHTTP_NO_REQUEST_DATA, 0, LDataLength, 0);
  if not LRes then
  begin
    LastError := GetLastError; // this returns 1209
    case LastError of
      ERROR_WINHTTP_SECURE_FAILURE:
        Exit(TExecutionResult.ServerCertificateInvalid);
      ERROR_WINHTTP_CLIENT_AUTH_CERT_NEEDED:
        Exit(TExecutionResult.ClientCertificateNeeded);
      else
        raise ENetHTTPClientException.CreateResFmt(@SNetHttpClientSendError, [GetLastError, SysErrorMessage(GetLastError, GLib.Handle)]);
    end;
  end;

ENetHTTPClientException is raised and captured in REST.Client

procedure TCustomRESTRequest.Execute;
      try
.....     

      except
        // any kind of server/protocol error
        on E: EHTTPProtocolException do
        begin
          FExecutionPerformance.ExecutionDone;
          // we keep measuring only for protocal errors, i.e. where the server actually anwered, not for other exceptions.
          LContent := E.ErrorMessage; // Full error description

          //Fill RESTResponse with actual response data - error handler might want to access it
          ProcessResponse(LURL, LResponseStream, LContent);

          if (E.ErrorCode >= 500) and Client.RaiseExceptionOn500 then
            raise ERESTException.Create(E.Message);
          HandleEvent(DoHTTPProtocolError);
        end;
        // Unknown error, might even be on the client side. raise it!
        on E: Exception do
        begin
          // If Execute raises an Exception, then the developer should have look into the actual BaseException
          raise ERESTException.CreateFmt(sRESTRequestFailed, [E.Message]);
        end;
      end;