There is nothing wrong with leaving the mosque after the adhaan to go and pray in another mosque

It is haraam to leave the
mosque after the adhaan without an excuse, because Muslim (655) narrated
that Abu’l-Sha’tha’ said: We were sitting in the mosque with Abu Hurayrah
when the muezzin gave the adhaan. A man stood up and walked out of the
mosque, and Abu Hurayrah followed him with his gaze until he exited the
mosque. Abu Hurayrah said: This man has disobeyed Abu’l-Qaasim (peace
and blessings of Allaah be upon him). 

It was also narrated by
al-Tirmidhi, who commented: Based on this, according to the scholars among
the companions of the Prophet (peace and blessings of Allaah be upon
him) and those who came after them, one should not leave the mosque after
the adhaan, except for one who has an excuse, such as not having wudoo or
something that cannot be avoided. End quote. 

Ahmad (10946) narrated it
with the addition: Then he said: The Messenger of Allaah (peace and
blessings of Allaah be upon him) commanded us: “If you are in the mosque and
the call to prayer is given, none of you should leave until he prays.”
Classed as saheeh by Shu’ayb al-Arna’oot in Tahqeeq al-Musnad. 

Excuses which make it
permissible to go out include: going out to do wudoo’ if the place for doing
wudoo’ is outside the mosque; going out with the intention of coming back,
such as if a person goes out to wake up his family, for example, then come
back. This applies to going out to pray in another mosque, if he knows that
he will be able to catch up with the prayer there. 

It says in Ghaayat
al-Muntaha: It is haraam to leave the mosque after the adhaan and before
praying without an excuse or the intention of coming back. … And it is
permissible to go out if one goes out after the adhaan but it is in order to
pray with the congregation in another mosque, especially if its imam is
better. End quote from al-Ghaayah ma’a Mataalib Ooli al-Nuha
(1/304). 

Shaykh Muhammad ibn
Ibraaheem (may Allaah have mercy on him) said in al-Fataawa (2/104):
The prohibition on going out of the mosque is subject to further discussion:
if there is no need for it and it serves no valid purpose, then it is haraam
to go out. That is because he looks like one who is going out of the mosque
without praying. But if he intends to pray in another mosque, or he has an
excuse or he intends to return and there is sufficient time, then it is not
haraam. End quote. 

Shaykh Ibn ‘Uthaymeen (may
Allaah have mercy on him) was asked: What is the ruling on a man doing
wudoo’ in one mosque then praying in another mosque? 

He replied: If the place
for doing wudoo’ is outside the mosque, there is nothing wrong with him
doing wudoo’ in this place and praying in another mosque, unless the iqaamah
has already been given in this mosque, in which case it is better for him to
pray there. If the place for doing wudoo’ is inside the mosque, then it is
not permissible for him to leave the mosque after the adhaan unless it is
for a reason, because Abu Hurayrah saw a man leave the mosque after the
adhaan and he said: This man has disobeyed Abu’l-Qaasim (peace and
blessings of Allaah be upon him). But if there is a reason, such as if he
wants to go to another mosque to attend a lesson or for some essential
purpose, then there is nothing wrong with him leaving, even if the place for
doing wudoo’ is inside the mosque. End quote. 

He was also asked: I see
some brothers sitting in the mosque for a lesson, then when the adhaan
begins, they leave. I have read that the Messenger (peace and blessings
of Allaah be upon him) forbade leaving the mosque when the adhaan is given.
What is the ruling on that? 

He replied: In al-Saheeh
it is narrated from Abu Hurayrah (may Allaah be pleased with him) that he
saw a man leave after the adhaan and he said: This man has disobeyed
Abu’l-Qaasim (peace and blessings of Allaah be upon him). The scholars
said: It is haraam to leave the mosque after the adhaan without an excuse.
But this hadeeth does not clearly state that the man went out to pray in
another mosque. He may have gone out so as not to pray. What we think is
that if you go out to pray in another mosque where you are sure that you
will catch up with the prayer, there is nothing wrong with it, but that
should not be done lest others follow that example and go out without
praying.

End quote from Liqa’
al-baab il-Maftooh (9/38). 

Hence you will know that it
is permissible for you to go out of the mosque after the adhaan in order to
pray in another mosque where the imam recites better or to meet some of your
brothers there, if you know that when you go there you will be able to catch
up with the congregation. 

And Allaah knows best.