Renewal before claim completed, does it impact service?

pjapk

We had an incident in May whereby an Asda delivery driver badly damaged our fence. They have admitted liability & that they won't contest a claim etc. so that's under way, but from what I've been told it can sometimes take a while for your own insurer to recover their costs from the 3rd party. In the meantime our renewal is due and an already fairly expensive policy (Legal & General) is 25% higher than last year, from £293 to £365. I don't know yet whether this is due to the claim, but I've searched online and have mentioned the claim in all quotes, but still have  at least 3 quotes below £200 and at least 13 quotes below £250.
I've emailed the current provider with the above and asked for an explanation, but if I jump ship (policy due 5th July) then is that likely to scupper any follow-up contact with the current provider to ensure they get all costs recovered (and my excess paid back)?