Traditionally, the likes of supermarkets tend to deny any liability or responsibility and string it out for as long as possible, unless there is overwhelming evidence against them.
if you were to approach the supermarket directly, they may make an offer, but this will really be peace money and will no doubt seriously undervalue your claim and the nature of your injuries.
Given that the injury only occured a week or so ago, I would be inclined to see if it settles down over the next couple of weeks, and if you still have problems, then consider making a claim, but for the full value rather than the pittance that the supermarket would offer you, if they made an offer at all.
A proper personal injury solicitor will be able to get the ball rolling under a no win no fee agreement (or under legal expenses if you have that) and will be able to give you a proper idea of the value of your claim, bearing in mind that you also have a history of slipped disc problems, the long term prognosis also has to be taken into consideration.
Whatever you do, don't use any of the claims management firms that stalk these boards (or any other claims management firms for that matter) they will rip you off as soon as look at you, go to a professional and local personal injury solicitor.
