Praise be to Allah.
Intention for making up missed fasts
It is not valid to change the intention of observing the voluntary fast that one has completed in order to make it a fast to make up for a day of Ramadan which one did not fast. When making up missed fasts, it is essential to form the intention the night before . Making up missed fasts comes under the same ruling as the original fast.
The Prophet (blessings and peace of Allah be upon him) said: “Whoever does not intend to fast before dawn, his fast does not count.” (Narrated by at-Tirmidhi, 730; classed as sahih by al-Albani in Sahih at-Tirmidhi)
At-Tirmidhi said after quoting it:
“What is meant, according to the scholars, is that the fast does not count for the one who did not form the intention to fast before dawn broke in Ramadan or when making up a missed Ramadan fast or when fasting in fulfilment of a vow. If he did not form the intention to fast from the night before, then it does not count. As for voluntary fasts, it is permissible for him to form the intention after dawn breaks. This is the view of ash-Shafi`i, Ahmad and Ishaq.”
An-Nawawi (may Allah have mercy on him) said:
“A Ramadan fast or making up a fast or an expiatory fast or a fast offered as fidyat al-Hajj (compensation for mistakes in Hajj) or other obligatory fasts are not valid if the intention is made during the day. There is no difference of opinion on this matter.” (Al-Majmu‘, 6/289) (See: al-Mughni by Ibn Qudamah, 3/26)
That is because changing the intention after finishing an act of worship does not have any impact.
As-Suyuti (may Allah have mercy on him) said in al-Ashbah wa’n-Nazhair, p. 37:
“If a person intends to end his prayer after completing it, that does not render it invalid, according to scholarly consensus. The same applies to all other acts of worship.”
So the fast that was done with the intention of observing a voluntary fast cannot count as a fast observed to make up for (a missed obligatory fast).
If he started observing it as a voluntary fast, then he decided during the day to change it into a fast observed to make up for (a missed obligatory fast), then he would have fasted part of that obligatory day on the basis that it was voluntary, so it cannot make up for an obligatory fast, because actions are but by intentions, and he fasted part of the day with the intention of it being voluntary.
And because he changed the intention from a general fast to a specific fast, and that is not valid.
Can you offer a voluntary fast before making up missed fasts?
We should point out to you that observing nafl fasts is not disallowed for the one who still has to make up Ramadan fasts , as mentioned in the question. Rather the correct opinion is that if a person wants to observe a voluntary fast when he owes an obligatory fast , such as making up missed days from Ramadan and the like, his fast is valid so long as he still has enough time to make up what he owes before the next Ramadan begins. But what is not allowed is to fast the six days of Shawwal before making up what he owes from Ramadan. However this is a matter concerning which the scholars differed.