To insert a split infinitive in the sentence "Day by day he went to hunt for wild-fowl in the woods; and it chanced that once he stood in the shores of a large lake," you could modify the adverb "to hunt" by inserting the adverb "to occasionally hunt" between the infinitive marker "to" and the verb "hunt." This would create a split infinitive, which is a grammatical construction where an adverb is placed between the infinitive marker "to" and the verb.
The modified sentence with a split infinitive would be: "Day by day he went to occasionally hunt for wild-fowl in the woods; and it chanced that once he stood in the shores of a large lake."