Normally claims are made against the policy that was in force at the time the loss event actually happened, in this case when the legal dispute can be proved to have started.
Yes you can seek a new policy with the same legal expenses insurers. Not sure this would make any difference.
You seem to misunderstand the way these legal expenses insurances work and I think you would find it very helpful, if you actually phoned the legal expenses number in the original policy to discuss this matter.
What you should be totally aware of, is that the legal expenses policies sold with Home Insurances, quite often do not provide any actual help. They will find reasons why they won't actually cover any legal costs. And having dealt with Insurance customers for a long time, one reason I often came across for declining legal costs, was a claimant delaying reporting the legal dispute they wanted help with.