"These are the verses of Allah which We recite to you in truth. Then in what statement (hadith in Arabic) after Allah and His verses will they believe?" (45:6)
There are several problems with your understanding of this verse.
First, we can concentrate on the meaning of "hadith." Hadith means speech or statement. However, in Islamic contexts, it has come to mean hadith specifically of the Prophet (SAW). So, the verse is rhetorically asking "What other speech will you believe in after Allah and his verses."
But, it is not referring to the hadith of the Prophet (SAW) specifically nor any other hadith specifically either. And it is clear that it does not mean to be comprehensive in what is allowed to believe. For example, several places in the Quran tell us to believe in the angels. However, that is another hadith to believe in "after Allah and his verses." Is that forbidden by this verse? No, clearly not because the verse does not mean to be comprehensive in what can be believed.
More obviously, the verse is rhetorical. The "after" in the verse is meant figuratively not literally. This can be shown by the fact that neither the verses before or the verses after actually refer to people who believe in the Quran and then believe in something else. All of the verses are referring to people who don't believe in the Quran in the first place [italics my comments explaining]:
"These are the verses of Allah which We recite to you in truth." [...] (45:6) How does the Quran being true argue against those who believe in the hadith of others in addition to the Quran? They already believe the Quran is true. You would expect something saying "The Quran should be alone or the Quran is completely detailed if that was the intent.
"Who hears the verses of Allah recited to him, then persists arrogantly as if he had not heard them. So give him tidings of a painful punishment." (45:8) Clearly referring to a person who rejects Allah's verses completely like the Kuffar of Makkah.
"And when he knows anything of Our verses, he takes them in ridicule. Those will have a humiliating punishment" (45:9)
"This [Qur'an] is guidance. And those who have disbelieved in the verses of their Lord will have a painful punishment of foul nature." (45:11) Again referring to disbelief rather than belief in something else.
So, the question is meant similar to "What thing are you going to believe after [foregoing] Allah and his verses?" And it is not meant to forbid believing anything else. Because if so, you would be forbidden from reading and believing any scholar. You would be forbidden from reading and believing any science. You would even be forbidden from believing this answer! But, that makes no sense. Clearly, you can find true things outside of Allah's verses and you can believe them. This verse is only a rhetorical question to those who disbelieve in Allah's verse on how they find anything else worth believing.
"And there is no creature on [or within] the earth or bird that flies with its wings except [that they are] communities like you. We have not neglected in the Register a thing. Then unto their Lord they will be gathered." (6:38)
This is not referring to the Quran and that is trivial to demonstrate. I can find a bird right now flying with its wings that is not mentioned in the Quran. This is referring instead to Allah's record and knowledge of everything. A similar verse mentioning this "register" or book is the following:
"And there is no creature on earth but that upon Allah is its provision, and He knows its place of dwelling and place of storage. All is in a clear register." (6:11)
"And We had certainly brought them a Book which We detailed by knowledge - as guidance and mercy to a people who believe" (7:52)
The question here is not whether the Quran is detailed with knowledge to be a guidance because everyone agrees to that. But, nowhere in the Quran does it say that the Prophet is not a source of guidance. In fact, repeatedly, Allah mentions the Prophet (SAW) as the second most important belief after Himself.
The Quran is clear on the matter of obeying the Prophet (SAW). The Quran says:
"He who obeys the Messenger has obeyed Allah ; but those who turn away - We have not sent you over them as a guardian." (4:80)
What is made clear is that when the messenger explains something or proscribes something as part of the religion it is a part of revelation from God, since God taught the messenger.