In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- There is nothing wrong with making up a missed Ramadan fasts in the second half of Shaban. This is not included in the prohibition of the Prophet (peace be upon him) on fasting after halfway through Shaban.
Answering your question, the Fatwa Center at Islam Q and A, states:
It was proven that the Prophet (peace be upon him) said: “When Shaban is halfway through, do not fast.” (Abu Dawud and Ibn Hiban)
There are some exceptions from this prohibition:
The evidence for that is the words of the Prophet (peace be upon him), “Do not anticipate Ramadan by fasting one or two days before it, except a man who fasts regularly, who should observe his usual fast.” (Al-Bukhari and Muslim)
2 – A person who started fasting before halfway through Shaban, and connects what comes after the halfway point to what came before. This is not included in the prohibition either.
The evidence for that is the words of Aishah (may Allah be pleased with her) who said: “The Messenger of Allah (peace be upon him) used to fast all of Shaban and fast all of Shaban except a little.” (Al-Bukhari and Muslim)
Al-Nawawi said: “He used to fast all of Shaban and fast all of Shaban except a little.” The second phrase is an explanation of the first, pointing out that by “all” what is meant is “most”.
This hadith indicates that it is permissible to fast after halfway through Shaban, but only for one who joins that to what came before the halfway point.
3 – An exception from this prohibition is also made for one who is making up missed Ramadan fasts.
Al-Nawawi (may Allah have mercy on him) said in al-Majmu`, 6:399:
“Our companions said: it is not correct to fast on the “day of doubt” just before Ramadan, and there is no difference of scholarly opinion on this point … But if a person fasts it to make up a missed day or to fulfil a vow, or as an expiation, that is acceptable, because if it is permissible to observe a voluntary fast on that day, it is more likely to be permissible to observe an obligatory fast… and if a person has to make up a day from Ramadan, then he has to fast it, because the time left for him to make it up has become very short.
The “day of doubt” is the thirtieth of Shaban if it has not been possible to sight the moon of the thirtieth because of cloud, fog, etc. It is called the “day of doubt” because there is some doubt concerning it – is it the last day of Shaban or the first day of Ramadan?”
Almighty Allah knows best.