Let us assert that the probability of the coin flip being heads is 1/2.
Now, you have awoken. Regardless of if you've awoken to the Heads flip, or the first time to the Tails flip, or the second time to the Tails flip, the original flip's chance was still 1/2.
The possible misunderstanding comes from the fact that you will awake TWICE to Tails, which is more than to Heads! But the thing is this is irrelevant to the question, because you don't know whether this is the first or the second time, and you only need to wonder whether the original flip (recall it is probability 1/2) was heads or not. Potentially you could think that you'd be "wrong" more often, but we are only looking at one specific instance of you waking up in isolation, for which you have no additional information.
For example, consider waking up 1000 times if the flip is tails. Upon waking up, do you think the probability heads becomes 1/1001? I think regardless of waking up the first time or the 1000th time, you have no more information so it might as well have been the first time, and hence the probability is 1/2.