Yes, it is mustahabb to separate between the obligatory and naafil prayers by speaking or moving to another spot.
The best way of doing so is by moving to offer the naafil prayer at home, because the best prayer a man can offer is a prayer that he offers at home, apart from the obligatory prayers, as was narrated in a saheeh report from the Prophet (blessings and peace of Allah be upon him). The evidence for the separation mentioned is the report narrated by Muslim in his Saheeh (1463) from Mu‘aawiyah (may Allah be pleased with him) who said: When you have prayed Jumu‘ah, do not follow it immediately with another prayer until you speak or leave (the mosque), because the Messenger of Allah (blessings and peace of Allah be upon him) instructed us to do that, not to follow one prayer immediately with another until we spoke or left (the mosque).
An-Nawawi (may Allah have mercy on him) said in his commentary on Saheeh Muslim:
This offers proof for the view of our companions – i.e., the Shaafa‘i fuqaha’ – that in the case of naafil prayer, both those that are offered regularly (Sunnah prayers) and others, it is mustahabb to move from where one offered the obligatory prayer to another spot, and the best is to move to one’s home; otherwise one may move to another spot in the mosque or elsewhere, so as to increase the number of places in which one prostrates, and so as to separate the naafil prayer from the obligatory one. The words “until we spoke” indicate that separating the prayers may also be done by speaking, but doing it by moving is preferable, because of what we have mentioned. And Allah knows best.
Abu Dawood (854) and Ibn Maajah (1417) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “Is any one of you incapable, when he prays, of stepping forwards or backwards, or to his right or left?” – meaning in order to offer a naafil prayer, i.e., a naafil prayer after an obligatory prayer.
Classed as saheeh by al-Albaani in Saheeh Sunan Ibn Maajah.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in al-Fataawa al-Kubra (2/359): The Sunnah is to separate obligatory and naafil prayers, in the case of Jumu‘ah and otherwise, as it is proven in as-Saheeh that the Prophet (blessings and peace of Allah be upon him) told people not to follow one prayer immediately with another until they separated them by moving or speaking. So one should not do what many people do, when they follow the salaam immediately with two rak‘ahs of Sunnah prayer, because this is going against the prohibition of the Prophet (blessings and peace of Allah be upon him). The wisdom behind that is to distinguish between that which is obligatory and that which is not obligatory, and also to differentiate between that which is worship and that which is not worship. Hence it is mustahabb to hasten to break the fast and to delay sahoor, and to eat before the prayer on the day of Eid al-Fitr, and it is disallowed to anticipate Ramadan by fasting one or two days before it. All of that is in order to separate between what is enjoined of fasting and what is not enjoined, and to separate between the act of worship and other acts, and also to distinguish between Jumu‘ah, which Allah has made obligatory, and other prayers. End quote.
By separating the obligatory and naafil prayers, one is distinguishing one of them from the other. Some of the scholars mentioned another reason for that, which is to increase the number of places where one prostrates, so that they will testify in one’s favour on the Day of Resurrection, as mentioned above in the words of an-Nawawi (may Allah have mercy on him).
Ar-Ramli said in Nihaayat al-Muhtaaj (1/552): It is Sunnah to move in order to offer a naafil or obligatory prayer from the place where one offered an obligatory or naafil prayer, so as to increase the number of places where one prostrates, for they will testify in one’s favour, and because that means filling more spots with worship. And if one does not move to another spot, then one should separate the prayers by speaking to someone. End quote.
And Allah knows best.