I would like to use some sprays as treatment for hair loss. The problem is that alcohol is an ingredient in these sprays. You previously stated that alcohol is impure although there is no evidence but the verse in which Allaah Almighty says (what means): {O you who have believed, indeed, wine, gambling, [sacrificing on] stone altars [to other than Allaah], and divining arrows are but impurity from the work of Satan, so avoid it that you may be successful.} [Quran 5: 90] Meanwhile, it is known that impurity could be tangible or intangible and that the two meanings are possible unless there is evidence that restricts it to a certain meaning. By consensus, gambling, [sacrificing on] stone altars [to other than Allaah], and divining arrows are an intangible impurity. There is also proof indicating that alcohol is an intangible impurity as well. It is narrated that Abu Hurayrah, may Allaah be pleased with him, said:
I knew that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, used to keep fast. I waited for the day when he did not fast to present to him wine that I made in a pumpkin. I then brought it to him while it fermented. He said, "Throw it to this wall, for this is the drink of the one who does not believe in Allaah and the Last Day."
There are other hadiths where the Prophet, sallallaahu ‘alayhi wa sallam, ordered shedding wine but did not order washing the vessels as he did with the pots which contained the flesh of wild asses, as he ordered either washing or breaking them. One of the principles of Usool Fiqh (fundamentals of Islamic Jurisprudence) is that giving up proclamation when needed is impermissible… This is the opinion of Ibn ‘Uthaymeen, may Allaah have mercy upon him, and others. What is your answer to this?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The majority of scholars are of the view, or rather narrated unanimously, that alcohol is impure. As evidence, they quoted the verse in which Allaah Almighty says (what means): {O you who have believed, indeed, wine, gambling, [sacrificing on] stone altars [to other than Allaah], and divining arrows are but impurity from the work of Satan, so avoid it that you may be successful.} [Quran 5: 90]

The impurity of alcohol, as the verse declares, is comprehensive and means both tangible and intangible impurity. According to the scholars of Usool Al-Fiqh, the word is interpreted with all the possible meanings. Here it is possible to judge alcohol as a tangible and intangible impurity. It is a tangible impurity in the real meaning and an intangible impurity in the metaphoric meaning. In this respect, it is similar to divining arrows as it is metaphorically impure. Using the joint word with its two meanings is the opinion of the early and late competent researching scholars of Usool Al-Fiqh. Ash-Shaafi‘i is one of the early scholars who stated this.

Quoting the Hadeeth mentioned in the question as evidence that alcohol is not impure because the Prophet, sallallaahu ‘alayhi wa sallam, did not order Abu Hurayrah to wash the vessel which contained it is invalid. That is because the Companions were formerly acquainted with the ruling that washing the vessels of wine is obligatory. They knew that washing all types of impurities in general, and what alcohol came in contact with in particular, was obligatory. It is narrated that Abu Tha‘labah asked the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, "We live with the People of the Book and they cook the flesh of swine in their pots and drink wine in their vessels (what should we do?)" The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, replied: "If there are others, then use them for eating and drinking. If there are no others, then wash them with water and then use them for eating and drinking." [Ahmad and Abu Daawood. This is the wording of Abu Daawood]

Consequently, to say that the Prophet, sallallaahu ‘alayhi wa sallam, did not order Abu Hurayrah her them, to wash the vessel which contained wine and that giving up proclamation when needed is impermissible is not deducible because the obligation of washing was known and therefore proclamation was not needed.

Allaah Knows best.