Apparently, this information is not essential for us to understand the Torah's message. While speculations as to the wood gatherer's identity are interesting, they are ultimately beside the point. This question can be asked in two ways. First, even if we assume that the Torah's stories occur in chronological order in this instance, there is a period of about 38 years in the wilderness, following the decree, in which there are only a few stories. The latter approach is the one taken in the Sifre adopted by Ibn Ezra and assumed by Rashi.
Scripture speaks in Israel's disgrace.
They only kept the first Sabbath, and the second one they profaned. Taking this position would arouse the question of why the Torah reported the story at this point if it happened in the first year. This defiance is a reference to the verses immediately preceding the story of the wood gatherer. The story was placed here, following the verses of the one who acts defiantly, as a real-life example of a person who acted defiantly and was punished with great severity. Assuming this position, namely that the incident of the wood gatherer happened in the second Shabbat after they left Egypt, would have the benefit of explaining why Moshe did not yet know that the Shabbat violator is condemned to death.