My q is this...can I use the current HDMI cable on the 4k 3rd Gen ?
You can use older types of HDMI cables, but it will likely limit what the devices and TV are capable of, when you get a newer TV as well. You are advised to get a new HDMI cable rated for 48 Gbps of bandwidth (“Ultra High Speed”), for use with current models HDMI devices and TVs with HDMI 2.1 ports. You don’t want the cable to be the reason for getting lesser quality video (resolution, dynamic range, frame rate).
It will be alright with HD 1080p non-HDR TVs, though. So, you don’t have to get a new cable until you get a new TV.
Set up your Apple TV - Apple Support
“On Apple TV 4K, you need an HDMI 2.0 or later compatible cable. To access all of the features that your Apple TV has to offer, use an HDMI 2.1 cable.”