A Google Books search for "always a bigger fish" yields two relevant matches, albeit with quite different connotations.
From "Where Ignorance Is Bliss," a letter to the editor of The Spectator, dated October 29, 1929 [combined snippets]:
... their lives and meet the Debit Men every Saturday night? Are they minnows on which to feed? If they are, I would remind you to look out since there is always a bigger fish around the bend (and he probably is a sucker, even so).
Here, the expression—although used figuratively—takes the perspective of a fisherman trying to catch a fish, and the general sense of the expression is optimistic: if you don't catch a fish here, you'll have a chance to catch an even bigger one at the next point along the river where you cast your line.
And from an unidentified article in The Fisherman, volume 6 (1955) [combined snippets]:
The average five-pound largemouth or two-pound smallmouth, for example, has no natural enemies to speak of, and he doesn't have to stay in shape to stay alive. This is why the biggest bass are often disappointing fighters. With salt-water fish it's a different story. No matter how big a jack, snook, or striper grows, there's always a bigger fish who'd like to make a meal of him, ...
Here the expression is literally about fish and makes the familiar observation that even a large fish is subject to being eaten by a larger fish.
Earlier than either of the two preceding instances is this instance that a Hathi Trust search turns up, from "British Sea Anglers' Society: Seventeenth Annual Dinner," in The Fishing Gazette (March 12, 1910):
Of many sportsmen it may be said that if you meet two of them, let them alone. It is true of golfers—let them alone, but not sea fishermen ; it is good to foregather with them, and always interesting, and there is always a bigger fish in the sea than has been taken out of it.
In this context, the expression, again used literally, indicates essentially that "there are more and better from where the current catch came from."
Of the two senses of the expression the one used literally in the 1955 Fisherman article seems to capture the idea of smaller fish being pursued by bigger fish in an endless pattern of larger and larger prey and predators.
Of course, the image of a very small fish being chased by a slightly larger fish, which is being chased by a slightly larger one, and so on, is a standard visual representation that seems to have been around for many years. Steven Pulimood, "Dreams of a Distant Wildness: Pieter Breughel," in the [New York City] Columbia [University] Spectator (October 3, 2001) reproduces a photo of a painting by Pieter Breughel the Elder called Big Fish Eat Little Fish (1556) that shows a monstrous fish being cut up by two fishermen and disgorging somewhat smaller fish that are themselves shown with fish in their mouth. The caption reads as follows:
In "Fish Eat Little Fish," Breughel was inspired by Hieronymous Bosch to portray proverbs in his work.
Whether the saying "Big fish eat little fish" was a proverb in Breughel's day or not, the observation has been a familiar one for centuries—and it is only one step beyond the commonsense figurative wisdom of that saying to note that, however big a fish may be, there is always a bigger one somewhere.