A faction of the Labour left led by Tony Benn and including the young Corbyn always opposed the EEC as a capiltalist plot. Wilson's referendum was his way of ending that division. He allowed cabinet ministers to campaign for either side. I guess much the same trick that David Cameron tried to pull except that he got away with it.
Don't know about a public clamour but the UK had been trying to join the EEC for about a decade, famously rebuffed by Charles de Gaulle, qui a dit "NON!".