It depends on whether the employer received your P60 in time and processed it. It sometimes takes until your second salary payment before any tax adjustments are made.
I suppose if you haven't worked for 8 months then you won't have earned anything this tax year so you won't have paid any tax this tax year before starting this job. So there is no rebate to give.
If the employer knows this is your first set of earnings this tax year your tax deduction will reflect this