Ultrastar TR-1 or similar Tranponder Bypass

Good day every one. Recently after a hours of installation of my first car alarm learned a lot about them. Alarm I have is a genuine ebay push button with a fob. It is cheap and suits none of my purposes, but one. I want to keep push start wiering and button with a OEM alarm I already have. I have 2010 Honda Accord Coupe 4Cylinder. Here is how I see it. 1. I have a OEM keyfob with which I open doors windows trunk and so on. 2. I have button installed in the car and it can start the car only when I apply brake. 3. When I sit in my seat car(TR-1 Bypass senses my key fob and car starts without shutting of. Here what I have already done. 1. I have button already wired up to the car. When I apply brake it will start it just with one click. (Key is not in the ignition) it will shut off the car because keyhole doesn't recognize the transponder. 2. I have the Ultrastar TR-1 unit which is wired to the car and if I put my key inside of the gray antena car will sense and start with push button without shutting off. (Key goes inside of the box in the gray receiver.) Antena from the TR-1 goes to ignition key hole. (Wires on the left side.) What do I need? I want to keep a key in my pocket and be able to start the car with push start button. How can I do it? Well here is the plan and my question. I want to spend how use this box as my new transponder key receiver redifecting it to the cars keyhole using antena from picture above. I hide the box under my seat and when it senses my key it activates and tells car that key fob is here with transponder. I use push button to start. Problem I have. Ultrastar TR-1 has a really small distance of transponder sense of have to be very close. I want to some how amplifie that signal. Not far at all, enough so when I sit down it reaches from my pocket to the under the seat. About 1 ft max. How can I amplifie that rfid signal from TR-1?????? I was thinking about adding a lot of copper wiering to the grey part this creating a bigger single. I can be wrong. Noob after all. If it possible which i am sure is.. well then I have stock alarm and a push start button which works only when my OEM key is with me.Boris Fua