I would reach the same conclusion as DaleM, i.e. that the arbitration clause still applies to the dispute in the question, but for a different reason.<1>
While the severability principle is indeed a concept in arbitration law, I don't believe that is the relevant doctrine here.<2>
Termination Of Employment v. Termination Of An Employment Contract
What terminates when a period of employment ends is the employment itself, not the contract of employment. The contract remains effective as to rights and obligations arising during the course of the employment covered by the contract.
(This concept would also sometimes be described in terms along the lines of "the right to a remedy for the workplace injury and the right to have disputes related to that resolved in arbitration are vested rights" that are not modified when the contract term of employment ends.)
This concept isn't particular to the arbitration obligation.
For example, if contract of employment that did not contain an arbitration clause provided that the employee was paid $200 an hour for the first six months and $220 an hour for the second six months of a one year fixed term employment agreement, but the employer failed to adjust paychecks according after the first six months and continued paying the employee at $200 an hour instead (assume to avoid the issue of waiver that this wasn't readily apparent on the face of the paystubs provided to the employee), the employee could bring suit thirteen months after the employee's employment terminated for the $20 an hour not paid as agreed during the last six months of the contract even though the employment period had ended.
The contract still remains in force to govern the rights and obligations of the parties arising during the period of employment.
Examples Of Termination Of The Contract Itself
In contrast, sometimes one contract is replaced, even retroactively, with another contract, in what is called a "novation" of the original contract. This truly does terminate the old contract, so that only the replacement contract remains.
So, if, for example, the original employment contract contained an arbitration clause, but this was replaced by a new employment contract without an arbitration clause three months later (in the sample case, before the worker was injured<3>), then the arbitration clause would not apply because the contract, and not just the employment was terminated.
Similarly, support that the workplace injury was the second dispute between the employer and employee that had come up. The first was a dispute over the rate of pay received which was resolved by a settlement agreement negotiated by lawyers for the parties before either arbitration or litigation in court was commenced, which expressly terminated all rights, known and unknown, of the parties arising under the contract, and the second was the workplace injury for which the relationship of the injury to work was only discovered later on. In this case, the contract and not just the employment had been expressly terminated, and so the arbitration clause would not apply to the workplace injury dispute (which would be barred by the settlement agreement and which may or may not have had an arbitration clause of its own).
<1> At least assuming that the dispute would have been subject to arbitration if a dispute were litigated while the employee was still employed. There can be circumstances when a workplace injury is not subject to arbitration even if the employer and employee undoubtedly agreed to an arbitration clause that applies to the dispute in question. For example, workplace injuries arising from sexual assault are not subject to arbitration in the United States. See 9 U.S.C. §§ 401-402.
<2> The severability principle usually concerns a determination of the enforceability of an arbitration clause in the face of certain kinds of arguments that the entire contract as a whole is voidable. It also does not apply in cases where there is a dispute over whether any contract of any kind was formed in the first place. For example, if someone presented the court with a contract containing an arbitration clause and sought to compel arbitration, and the defense to a motion to compel arbitration was that the defendant had never met or had any connection with or communication with the defendant or anyone related to the alleged contract.
<3> The hard case, where the novation takes place after the injury, is complicated by legal doctrines regarding the conditions under which a vested legal right can be waived, which may or may not be met depending upon the circumstances under which the novation was entered into by the parties and presents conceptually distinct legal questions in addition to the legal questions already present in the simple case where the injury takes place after the novation.