Hi Squirrel
The only reason you would not need public liability would be if you were an employee of the salon (ie, they were paying you and deducting tax and national insurance). They may (should) have their own insurance, but if anything happens to a client which they consider to be as a result of your treatment, or happens while they are in your care, before, during or after the treatment, they're your responsibility and you need to be insured. Usually the cheapest way is to take the insurance offered by your professional body. You could have a look at the FHT website. They seem to be cheap, although you would, of course, have to weigh the cost up against what you get for your money.
hth
Lynne