How would a minimized program listen, but still allow keyboard events?

Question

The goal is to have a minimized application, that can respond to keyboard presses.
The keyboard presses will still be able to affect other applications in focus.

The only way I have found to detect keyboard presses within the minimized application,
is to use a low level keyboard hook:

using System;
using System.Diagnostics;
using System.Windows.Forms;
using System.Runtime.InteropServices;

class InterceptKeys
{
    private const int WH_KEYBOARD_LL = 13;
    private const int WM_KEYDOWN = 0x0100;
    private static LowLevelKeyboardProc _proc = HookCallback;
    private static IntPtr _hookID = IntPtr.Zero;

    public static void Main()
    {
        _hookID = SetHook(_proc);
        Application.Run();
        UnhookWindowsHookEx(_hookID);
    }

    private static IntPtr SetHook(LowLevelKeyboardProc proc)
    {
        using (Process curProcess = Process.GetCurrentProcess())
        using (ProcessModule curModule = curProcess.MainModule)
        {
            return SetWindowsHookEx(WH_KEYBOARD_LL, proc,
                GetModuleHandle(curModule.ModuleName), 0);
        }
    }

    private delegate IntPtr LowLevelKeyboardProc(
        int nCode, IntPtr wParam, IntPtr lParam);

    private static IntPtr HookCallback(
        int nCode, IntPtr wParam, IntPtr lParam)
    {
        if (nCode >= 0 && wParam == (IntPtr)WM_KEYDOWN)
        {
            int vkCode = Marshal.ReadInt32(lParam);
            Console.WriteLine((Keys)vkCode);
        }
        return CallNextHookEx(_hookID, nCode, wParam, lParam);
    }

    [DllImport("user32.dll", CharSet = CharSet.Auto, SetLastError = true)]
    private static extern IntPtr SetWindowsHookEx(int idHook,
        LowLevelKeyboardProc lpfn, IntPtr hMod, uint dwThreadId);

    [DllImport("user32.dll", CharSet = CharSet.Auto, SetLastError = true)]
    [return: MarshalAs(UnmanagedType.Bool)]
    private static extern bool UnhookWindowsHookEx(IntPtr hhk);

    [DllImport("user32.dll", CharSet = CharSet.Auto, SetLastError = true)]
    private static extern IntPtr CallNextHookEx(IntPtr hhk, int nCode,
        IntPtr wParam, IntPtr lParam);

    [DllImport("kernel32.dll", CharSet = CharSet.Auto, SetLastError = true)]
    private static extern IntPtr GetModuleHandle(string lpModuleName);
}

This works, but it prevents the key presses from having any effect on other windows in focus. I want the minimzied application to listen to these keyboard presses, but not have the hook be the only program who knows about the key press.

How can I listen for keyboard events, while still allowing other applications to see these keyboard events?

I thought there might be setting to allow LowLevelKeyboardProc to do this, but I have not found any.