Should he say the adhaan if he wants to pray on his own in the mosque?

The scholars are agreed that the one who enters the mosque
and prays alone does not have to say the adhaan or iqaamah, and that the
adhaan and iqaamah of the muezzin in the mosque are sufficient for him. 

Imam al-Shaafa’i (may Allaah have mercy on him) said in
al-Umm (1/106): 

I do not know of any disagreement with the view that if a
person comes to the mosque when the imam has finished praying, he may pray
without any adhaan or iqaamah. End quote. 

It is mustahabb for him to say the iqaamah before beginning
to pray, and it is best if he says the adhaan and iqaamah, because the
adhaan and iqaamah are dhikr or remembrance of Allaah. This idea was
indicated by Qataadah (may Allaah have mercy on him) when he said: Nothing
but good will come to you from testifying that there is no god but Allaah
and that Muhammad is the Messenger of Allaah. 

It is proven that Anas (may Allaah be pleased with him)
entered the mosque when they had finished praying, and he told a man to say
the adhaan and iqaamah. Narrated by al-Bukhaari in a mu’allaq report and by
Ibn Abi Shaybah in al-Musannaf (1/250). Classed as saheeh by
al-Albaani in Tamaam al-Minnah (p. 150). 

Sa’eed ibn al-Musayyab said concerning people who came to the
mosque when the prayer had ended: they should say the adhaan and iqaamah.
This is the view of Imam al-Shaafa’i (may Allaah have mercy on him). 

See: Mughni al-Muhtaaj (1/318) and al-Nawawi in
al-Majmoo’ (3/93). 

Ibn Qudaamah said in al-Mughni (2/74): It is better
for each worshipper to say the adhaan and iqaamah, unless he is making up a
missed prayer or at a time other than the time of the adhaan, in which case
he should not say it out loud. End quote. 

i.e., so that he will not confuse the people with his
adhaan. 

He also said (2/79): 

If a person enters the mosque where the prayer has already
been offered, if he wishes he may say the adhaan and iqaamah. This was
stated by Ahmad. 

And if he wishes he may pray without any adhaan or iqaamah,
because ‘Urwah said: If he comes to the mosque when the people have already
prayed and said the adhaan and iqaamah, then their adhaan and iqaamah will
suffice for those who come after them. This is the view of al-Hasan,
al-Sha’bi and al-Nakha’i, but al-Hasan said: It is better, in their view, to
say the iqaamah. 

If he says the adhaan it is mustahabb for him to say it
quietly and not out loud, so that the people will not be confused by an
adhaan that is given at a time other than the usual time. End quote. 

See also the answer to question no.
5660 and
6130. 

And Allaah knows best.