Is it permissible for a poor employee at a charity organization to take from the zakat designated for the poor, even if there are those in greater need than him?

The hadith mentioned in the question pertains to the grants of the ruler and the rights of Muslims from the public treasury (Bayt al-Mal), not to the funds of charity (sadaqat). As for the questioner, if he is poor or needy (his salary does not cover his needs, or he has permissible debts that he cannot repay), then it is permissible for him to take zakat. Preferring others who are in greater need is commendable.