Hello,
I completely agree with the other responses. Law is extremely competitive so don't overthink what size firm you go to. The name really is irrelevant, for work experience, the focus should be on the skills and experience rather than the name of the business. Also, you don't need to go and get lots of work experience in loads of different firms, two or three is enough. Also, part of it concerns where you want to go and what you want to do. For instance, local Barrister chambers don't want to give pupillage to people who just then want to go to the bigger firms, they want people who are keen to stay in that area.
Plus, if you want to be a lawyer you need to be able to work with people. By taking advantage of as many opportunities as you can, it means someone else will be missing out on those opportunities. There is obviously a need for some self-mindedness, but when applying for a job if you can demonstrate that you don't only care about yourself and your own improvement but are equally keen on helping those around you improve, even if they are opposing counsel, this will make you a much better person and much more respected in the legal industry.
I hope this helps, if you have any further questions please do let me know.
Charlie
Law LLB Student