You can't display it until the 1st of the month as you are committing an offence by not displaying a valid tax disc.
If your new disc starts 1st July and you display it from 20th June, you aren't displaying a valid disc (new one isn't actually valid yet) and you could get fined.
There is no legal "grace period" and you could be proscuted, in theory just for one day but in reality the authorities will start to take an interest aftter a week to 2 weeks.
You can apply to tax your vehicle from the fifteenth day of the month in which the existing vehicle tax is due to expire. The tax disc will come into force on the first day of the following month and should not be displayed on your vehicle until then.