I'll show you mine if you show me yours...

As a lay historian (at best), my reading list is pretty haphazard. I'll say that the Corrupting Sea does look great, so thank you for that!

For broad European context, I quite like Richard C Hoffman's An Environmental History of Medieval Europe. In addition to being well written and covering the basics from neolithic to early modern Europe, also has a very solid bibliography itself.

Possibly of specific interest to you, though I'm not informed enough to know if anyone would call them "Essential":

• Environmental Problems of the Greeks and Romans: Ecology in the Ancient Mediterranean by J Donald Hughes
• Squatriti, P. (2013). Landscape and Change in Early Medieval Italy: Chestnuts, Economy, and Culture. Cambridge: Cambridge University Press.
• Appuhn, K. (2000). Inventing Nature: Forests, Forestry, and State Power in Renaissance Venice. The Journal of Modern History,72(4), 861-889.

For broader context on Environmental History, there's a couple of newer books I'm excited about but haven't read:

• Environmental Thought: A Short History by Robin Attfield
• The Environment: A History of the Idea by Paul Warde, Libby Robin, and Sverker Sörlin

Then there's a lot of good books on the history of modern environmentalism, if that's of interest to you:

• John Muir and His Legacy: The American Conservation Movement by Stephen R. Fox,
• Forcing the Spring: The Transformation Of The American Environmental Movement by Robert Gottlieb
• Beloved Beasts: Fighting for Life in an Age of Extinction by Michelle Nijhuis

This isn't terribly thoughtful but I would also pick up one or two of the recent issues of the Journal of Environmental History. Modern environmental history has so many tools that weren't around even a decade ago, Environmental History (the journal) in my experience tends to have some of the best work on historical climate change, forestry and natural resource use, and environmental history outside of a European or American context.

A book I consider essential reading is Tending the Wild: Native American Knowledge and the Management of California's Natural Resources by Kat Anderson. Possibly less relevant if you don't happen to be in California, but it does contain a lot of fantastic work. It’s certainly informed my reading of others' work on both modern, prehistoric, ancient society's histories of environmental stewardship and resource management.

I'd love to hear back if you find any gems you're putting on your list, or get feedback from your advisor down the line. While I'm here, when I wrote this answer a few weeks ago I found myself wishing for a classicist who might have some more insight on what ancient authors were saying or not saying when they wrote about environmental problems, so if you go take a look and have any thoughts, definitely let me know!

William Cronon's 1983 Changes in the Land is very much a classic, it's on the environmental history of colonial New England and the differences between Native land use and Colonial land use. The 2003 re-issue had some useful updates, so that's the edition you want.

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