Seems you are SOL for the bypass harness... Even Metra doesn't have one.
Time to get your meter out, and start poking into plugs! Have you checked for voltage (power) at the amplifier's input? Here's where I would start looking.
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It all reminds me of something that Molière once said to Guy de Maupassant at a café in Vienna: "That's nice. You should write it down."
^Well, even though there is no harness, that doesn't mean it can't be done...right? Here's my idea:
^ Yes. But, even with the old head unit, the speakers wouldn't work.
To bypass the amp, this is my understanding of what to do: (please correct me if I'm wrong)
1. Cut each of the white connector ends off of the wire sets in order to gain access to the wires.
2. Following the wiring diagram below, splice the input wires with their corresponding output wires. The inputs are colored the same as their corresponding outputs. Since I have front tweeters, won't I just twist the output of those together with the output of the front door speakers and then splice those with the front input? (for example: there is only one right front input but two right front outputs; one for tweeter and one for door speaker.)
https://www.crutchfield.com/App/Support/KnowledgeBase.aspx?ifs=hyundai+tiburon
OUTPUT:
INPUT:
that looks about right but make the wires not in use are not bare. ex.... remote, power, and etc. leave them plugged in or tape the unused wires and enjoy!
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If I'm to loud, then you're too old!!!
Before you go and start cutting up the factory wiring why don't you confirm that it is the stock amp first? Do you have a DMM like haemphyst said?