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The phrase "a further layer of" is correct and usable in written English.
It can be used when discussing additional complexity, depth, or detail in a subject or concept.
Example: "The research provided a further layer of understanding regarding the impact of climate change on biodiversity."
Alternatives: "an additional layer of" or "a deeper layer of".
Exact(60)
There is a further layer of complexity to this.
To men case-hardened by work they added a further layer of political hardening.
That it is the NPG that is hosting the show adds a further layer of complexity.
Revealing how the mechanics work can also be a double bluff, a further layer of illusion.
Somaliland's financial dependence on remittances sent by those who work overseas adds a further layer of tension and mutual hostility.
The problem has a further layer of irony because – to simplify for a moment – GCHQ actually invented computers.
Now the doctor's namesake, Sebastian Barry, has added a further layer of confusion: burying him under a midden of adjectives.
Moves to protect depositors in bail-ins add a further layer of complexity.
These limits are designed to protect the breeding population, and they add a further layer of discrimination to what was already, by its nature, a fairly discriminating industry.
This is Mozartian, not because it echoes the music but because, audaciously yet judiciously, it provides a further layer of rhythm, phrasing and harmony.
To add a further layer of confusion, Mick Sullivan's tally of 41 in 46 Great Britain caps from 1954-63 remains well in the distance.
More suggestions(18)
a supplementary layer of
a complementary layer of
a further level of
a further depletion of
a further coat of
a greater layer of
a deeper layer of
a further gesture of
a further demonstration of
a further type of
a further implication of
a further prediction of
a further role of
a further order of
a further evolution of
a further decrease of
a further sign of
a further sense of
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com