It is mustahabb to separate between the obligatory prayer and the naafil prayer, by speaking or moving

Praise
be to Allah

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.