Praise be to Allah.
There is nothing wrong with a person joining a voluntary fast to a made-up fast or six days of Shawwaal, because of the general meaning of the evidence which encourages offering voluntary fasts without a separation between the voluntary fast and the made-up fast.
If what is meant by your continuing to fast after completing the six days of Shawwaal is observing voluntary fasts continually until the end of Shawwaal or for certain days that you want to fast voluntarily, seeking thereby the pleasure of Allah, there is nothing wrong with that, so long as neither of you will be harmed by that and neither of you will fail to fulfil the rights of the other.
However the best teaching concerning that is not to fast for an entire month except the month of Ramadan; rather the days when one fasts should be mixed with days when one does not fast, and vice versa, as was the practice of the Prophet (blessings and peace of Allah be upon him).
It was narrated that ‘Aa’ishah (may Allah be pleased with her) said: The Messenger of Allah (blessings and peace of Allah be upon him) used to fast until we said that he would not break the fast, and he used to refrain from fasting until we said that he would not fast. And I never saw the Messenger of Allah (blessings and peace of Allah be upon him) fast any month in full except Ramadan, and I never saw him fast more than he did in Sha‘baan.
Narrated by al-Bukhaari, 1969
Shaykh Ibn Jibreen (may Allah have mercy on him) said: It is permissible to fast continually on consecutive days, then not to fast continually on other consecutive days. The evidence for that is the hadeeth quoted in the question, because that is voluntary and mustahabb.
End quote from Fataawa ash-Shaykh Ibn Jibreen
But if what is meant is continuing to fast until the next year apart from the two days of Eid and the days of at-tashreeq (the three days following Eid al-Adha), this is what the scholars call lifetime fasting (sawm ad-dahr) and the ruling on it is that it is makrooh according to the correct scholarly opinion.
For more information, please see the answer to question no. 144592
And Allah knows best.