Your salary till the date of leaving your employment with the previous company should have been reflected in Form 16 issued by the employer, after deducting salary for the notice period. If it is not done so, you may, while filing your IT Return, offer only salary after deducting salary for the notice period.
In the case of new employer, whatever is paid will be salary including the reimbursement of your notice period pay.
Thus you will be paying taxes only on your actual salary earned and there is no extra tax to be paid by you.