what is his/her new consideration for the employer? Does the absence
of one render the pay rise unenforceable?
No, the pay rise is enforceable, especially where employment is at-will because either party is entitled to terminate employment anytime so as to prompt a more convenient rehire anyway. Law is not ritualistic, whence termination is not a legal requirement for the new conditions to be enforceable. Thus, @RonBeyer is right, although in his comment I would place emphasis on the aspect of employee's "continued" work rather than the "good" quality thereof.
Even if the initial contract binds the parties for a pre-established duration or perpetuity, nothing prohibits the parties to mutually agree to modify that contract. For instance, see the Restatement (Second) of Contracts at § 89 (listing alternative conditions so that "[a] promise modifying a duty under a contract not fully performed on either side is binding").
Prohibiting a mutual agreement merely because one of the parties does not add a novel consideration would contravene the contract law principle of parties' knowingly and willfully binding themselves to updated terms. In the scenario you describe, there is still a consideration (namely, the employee's work), and the pay rise would not materialize if the [employee's] consideration ceased to exist. See also Restatement at § 311(2).