Praise be to Allah, Lord of the Worlds.
Breaking one`s prayer (Praying alone) to join the congregation is a broad topic that covers a wide spectrum of cases and details over which jurists` have dwelled extensively, and they boil down to the following:
First: If he was performing Nafil prayer (Voluntary prayer), then there is no harm in breaking it to join any congregation, be the latter Fard (Obligatory) or Nafil. However, it is better that he doesn`t prolong the Nafil to be able to catch up with the congregation.
Second: If he was praying Fard, then he isn`t permitted to break it unless he turns it into Nafil prayer by changing his intention to that end while praying. Having done so, he is permitted to break his prayer as clarified in the above point. However, if he wants to know which is better, then he should consider the following details.
Third: If he was performing a two-Rakah-Fard prayer and didn`t fear missing the congregation, then it is recommended that he finishes the Fard first then joins the congregation.
Fourth: If he was performing a three or a four-Rakah-Fard prayer and stood up to deliver the third Rakah, then it is also recommended that he finishes his prayer and joins the congregation afterwards. Fifth: If he was performing a three or a four-Rakah-Fard prayer, but hasn`t yet stood to offer the third Rakah, then it is recommended that he changes his intention from offering Fard prayer into offering Nafil prayer, and suffice with two Rakahs, then join the congregation. This is unless he feared missing the congregation if he completed the two Rakahs. In this situation, it is better that he breaks the Nafil prayer and joins the congregation.
Sheikh Zakaria Al-Ansari (May Allah have mercy on him) said: "If the congregational prayer was established while the person was praying the Fajir
First: If he was performing Nafil prayer (Voluntary prayer), then there is no harm in breaking it to join any congregation, be the latter Fard (Obligatory) or Nafil. However, it is better that he doesn`t prolong the Nafil to be able to catch up with the congregation. However, if he wasn`t performing Fajir and hasn`t stood up to deliver the third Rakah, then he should intend it as Nafil prayer and suffice with two Rakahs, then join the congregation. Rather, if he feared missing the congregation in case he completed two Rakahs, then it is recommended that he breaks his prayer and joins the congregation." {Al-Majmou`}. Moreover, it is recommended that he break his prayer if he feared missing the congregation once the Imam makes Tasleem.
Indeed, if he was expecting another congregation to be established soon and there is ample time for that, then it is better that he finishes his Nafil prayer, then joins the new congregation from the very beginning." This excerpt was cited from {Asna Al-Matalib Shareh Rawd At-Talib, vol.1/pp.231}. And Allah Knows Best.