To what extent is the search for a lawful (halal) job strict, considering the presence of a suspicion of unlawfulness (haram) in some available jobs, such as delivering milk that may be used during the day in Ramadan, or transporting nurses' clothing that may reveal their 'awrah (parts of the body that should be covered)?

It is permissible to deliver milk to restaurants and markets, except during Ramadan (definition), if it is predominantly consumed during the day. In that case, it is considered aiding in a forbidden act, in accordance with the Almighty's saying: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). The majority of scholars hold that anything intended for forbidden purposes is forbidden, and disbelievers are bound by the subsidiary rulings of Islamic law. If the country has Muslims, and the purchase is not predominantly for breaking the fast during the day, then there is no harm in delivering it to the market, while avoiding restaurants that serve it during the day.

As for washing clothes, what matters is the majority; if the majority are for men or for women who do not display themselves immodestly, then there is no harm. You should seek forgiveness and send blessings upon the Prophet, peace be upon him, based on the Almighty's saying: (And I said, 'Seek forgiveness of your Lord. Indeed, He is ever Forgiving. He will send [rain from] the sky upon you in [copious] showers and will extend you with wealth and children'), and his, peace be upon him, saying to the one who dedicated all his prayers to him: (Then your worries will be sufficient for you, and your sins will be forgiven).