Well the simplest approach would be to upgrade your UNO to a Nano (8 analog inputs), Mega (16 analog inputs) or Leonardo (12 analog inputs).
Failing that, you would probably want to go to i2c or spi, which are serial protocols. You would issue a command to say which value you want to read, and then read that value.
I've not used this product, but this is the first product that came up via google for i2c analog input arduino, and it looks like it would give you 8 ports of 12-bit (0..4095) reads (a5/a4 would be used to read the i2c values): http://www.gravitech.us/i2c128anco.html. Combined with the other 4 inputs, this would give you 12 inputs.