If I wanted to say something occurred occurred between 1627 and 1629, I would use "the late 1620s". However, if I wanted to say something occurred between 1607 and 1609, I couldn't use "the late 1600s" because that would imply it occurred between 1670 and 1699.
There is no broadly well-understood way of expressing this unambiguously in the form you want. Some people use "aughts" to refer to the first decade of a century, as in "late sixteen-aughts", but I suspect that would be harder to parse for most people than just rephrasing as "occurred between 1605 and 1610". This is just a gap in the language.
Using "1600s" for the decade would stop the average reader in their tracks; I would interpret it as referring to the century. To clarify, a writer could take a few approaches:
1. Using parentheses
Using parentheses or other means, one might specify without being too explicit.
The "aughts" is one way of referring to the first decade of a century, in American English, such as 2000s (decade).
— Wikipedia (emphasis mine)
You could apply this as follows.
One of the first historic moments of the 1800s (the decade) was President John Adams moving into an unfinished White House.
2. Using a relatively new word
In appropriate context, you could use one of the newer words for "first decade of a century":
For example, one may write,
The 1800s in U.S. Politics
- President John Adams moved into the White House in the aughts.
These have an informal ring to them. We have only relatively recently started using these⁵, so they are discontinuous in regional popularity.
3. Being explicit
In my view, this extract states the best practice in this case, which I, as a reader, find concise enough.
When writing about the first two decades of a century, it’s probably best to be a little wordy for the sake of clarity. For example:
History seemed to repeat itself in the decade 2000–2009.
— Daily Writing Tips (emphasis mine)
Some might even prefer an even longer version:
Documented here is the first decade of the 1800's with duels, battles, explorations, and births in the U.S. and abroad.
— ThoughtCo (emphasis mine)