Just randomly looking at Arsenal's wikipedia entry, according to our honours list, we won some competition called Southern Professional Floodlit Cup in 1958/59, what a name We beat Crystal Palace in the final 2-1.
Apparently it involved clubs from London, South East England and a few clubs from the Midlands. The competition existed for just a few seasons in the 50s.
The competition gave way to what we now know as the League Cup.
Some more info/history about the competition here: http://www.blog.woolwicharsenal.co.uk/archives/11928
In the above link, there's an interesting quote regards to our approach to not wanting to enter the League Cup.
The reason for being part of the Floodlit Cup but not the League Cup seems to have been two fold. First, that the new cup was described by the founder Football League President Joe Richards, as an interim step towards league re-organisation – which Arsenal opposed. Second that the League Cup was open to clubs from across the country – whereas the Floodlit Cup was very much a London affair for most of the timee.
To give the context of this, there was much talk about declining crowds in the era, and while Arsenal v Tottenham in September 1960 got 60,088 into Highbury, other matches got far fewer. Arsenal v Fulham on 3 April 1961 attracted only 20,142 for example. Richards spoke of ‘cutting down the number of clubs in each division… and even given more consideration to the system of four up and four down’. Arsenal, a very conservative club at the time, wanted none of that.
Arsenal, a conservative club in 1960? I guess things don't really change