Interesting suite of responses so far but not much of any actual value.
Your question is based upon a somewhat flawed premise - "...can private action be taken by a councillor once employment has ceased?"
You are mixing apples, bananas and oranges there.
If there is an issue of defamation between 2 individuals it is a matter for those 2 individuals.
If the defendant is an employee (at the time of the alleged defamation) and the defamatory action was in the process of their duty / work, then there may be a corporate liability. That's why PCs have insurance.
Your question is flawed because the use of the 'Cllr' descriptor infers that they may be acting in an official, rather than a private, capacity.
So, a Cllr, would not seek a defamation claim but a private individual might. That individual would have to decide if they were going to progress a corporate claim against the PC or an individual claim against the former employee.
Tricky, but either way, you'd do well to familiarise yourself with the pre-action protocols (https://www.justice.gov.uk/courts/procedure-rules/civil/pdf/protocols/prot_def.pdf) as a first step, then seek professional legal advice, and then take a long hard think about the commonly applied mantra for such circumstances - the courts are full of righteous, but broke men. (I say men because it is usually men that suffer from the ego which requires them to progress a course of action with little chance of actual success regardless of their determination, often based upon strongly held principles, but even a slim chance of satisfaction is likely to leave them completely pot-less.)
I absolutely sympathise with the circumstance you describe having experienced very similar and with the knowledge that the miscreant employee that imposed their adverse influence in this PC had done EXACTLY the same in her previous employ and so yes, the principle of moving from PC to PC almost without accountability is a very real world situation which is exacerbated by poorly understood and inappropriately applied termination contracts with confidentiality clause.