Where in Genesis chapter two is there any chronological order presented, other than, what is demanded by the context of the chapter's narration?
The chapter begins by closing the discussion of chapter one and opening a new topic. The new topic is GENERATIONS of....
Which has nothing to do with chronological order as any requirement.
In chapter two there exists no terms implying chronology.
The word "AND" doesn't mean "AND THEN".
Chronology is implied by the Hebrew "waw-consecutive" or "preterite" construction, which runs through chapter 2 as well as chapter 1. This is the normal grammatical construction used for narrative. It implies a blow-by-blow chronological account. Thus it often
is translated "and then" or "then" instead of "and." Check some other translations for chapters 1 and 2. Quite a few of them (NASB, NLT, NKJV, HCSB, New Century, God's Word) sometimes render the waw-consecutive as "then" in these chapters.
But the waw-consecutive construction does not
always show chronology. E.g. Gen 2:15 starts with a waw-consecutive, but it is actually a restatement of Gen 2:8b. It does not indicate that man was placed in the garden twice. (Gen 2:9-14 is a non-chronological side comment; the waw-consecutive at Gen 2:15 repeats the last chronological piece and serves to pull us back into the chronological account.)
So the waw-consecutive construction in chapter 2
normally indicates chronology, but not always. These non-chronological exceptions should be determined from the literary context.
One could argue that the chronology of chapter 2 should defer to that presented in chapter 1, since chapter 1 came first. On the other hand, chapter 2 reads more like a historical, chronological narrative, with a similar style to the following chapters. Chapter 1 is different; it is much more stylized and formulaic in its construction. So one could also argue that the chronology in chapter 2 should take precedence over chapter 1 on this basis.