You have to read what the paperwork says from the court and / or DVLA.
If it says your ban ends on 23rd October then you can drive from the 24th.
if your thinking is: I was banned on the 23rd so it runs out on the 23rd, it actually finishes on the 22nd, so you can drive from the 23rd.
Either way, you can apply for your licence 2 months early (3 months if you are a High Risk Offender) and you should look at the date that is on the licence when it comes back. Whatever date that says is the date that you can start driving, you do not have to wait until the day after.
Hope that helps.