I don't understand why you feel the need to 'build' any antennas, at all.
Most likely, the receiver is suffering from a lack of ground plane. Expanding that will help a great deal.
However, and, as a general class ham, you should remember this, you really need a better transmitting antenna.
Now, I don't know what the law is on that particular unit, and I'm not motivated to refresh myself, but potentially you are violating some regulations tinkering with a permanently-mounted antenna on the transmitter.
I personally would get a 1/4 wave UHF antenna, or even a small yagi, have its' centerpoint adjusted to the tx frequency, and mount that on a pole at the sensor point. Maybe ten or fifteen feet even. Especially if the sensor is on the wrong side of the car (or person) it is detecting. I'd use the most efficient coax I could locate, and I'd make sure the antenna system was resonant to the transmitter.
At distance, and with shit antennas and lots of ground clutter, having the detected object in between the output antenna and the receiver location may be enough to make the system partially nonfunctional.
The center wire, as previously mentioned solders into where the wire rod exits the transmitter. The braided part of the coax should solder to ground, which will probably be on the back side of the board, and may have a conformal coating applied.
I'd test that, first. I'd also ground the antenna pole in case lightning is an issue.
If you're still having trouble, I would then have someone repeatedly trip the sensor, as you walk the receiver around in the space where it lives. Sometimes, just a few feet difference is all it needs - no modifications necessary. Up high, and as previously mentioned, the antennas all need to point in the same direction, preferably vertical (unless you bought the yagi - then it needs to point at the receiver).
Failing that, then I'd either look for a gain UHF walkie antenna, and mod a BNC bulkhead connector onto the receiver, or if I was determined to leave the receiver somewhere, I'd hook me an outside antenna onto the house, and run a coax back to the receiver.
Or, I'd move the receiver to where it reliably picked up the sensor, and then I'd tie something to the alarm output to warn me whereever I was. Example: most doorbells have two dingers, front door and back door. Almost no one utilizes the back door. Tie the output to pulse the back door tone in the door bell.
Lots of stuff you can do there