This jins is very similar to Jins Secondary Rast, as the top half of Maqam Ajam. Rather than tonicizing the note on which it starts, according to traditional theory--the 5th scale degree above Jins Ajam--it leads up, 5-6-7-8, to tonicize the octave. Even when descending, one feels the tonic pull from the octave. Again, the use of two numbers in identifying the Jins ("2nd Ajam 5/8") refers to the traditional tonic and starting point of the Jins (5), and the actual tonic, i.e. note of resolution of the Jins (8).
First, from the Muwashshah "Ayqadha-l-hubbu Fu'aadi" (#03 & #05):
From Lissa Fakir (#02):
From the Abdel-Wahhab Mawwal Rast "Kull illy Habb Itnasaf" (#32 & #34)
where Maqam Ajam starts on the 4th scale degree and stretches to the 11th scale degree--the Abdel-Wahhab Mawwal Zanjaran "Amana Ya Leil" (#40, #42 & #44):
where Jins Ajam frequently occurs on the 6th scale degree, so one can find Secondary Ajam leading up to that tonic 3-4-5-6. These examples, however, are much more ambiguous than the previous ones, because--as I discuss on the page Bayati Pathways: Nahawand & Ajam--there seems to me to be much more ambiguity in Bayati tonicization in the lower 6 notes, than elsewhere in the maqam system. So an argument could be made regarding some of these examples, that what I'm calling "Secondary Ajam 3/6" could just be called "Ajam 3," because one feels the tonic pull on the 3rd scale degree stronger than on the 6th in certain melodies... I leave it to you to decide how you feel about it...
First, from the famous wedding song "Ah Ya Hilu" (#05)
From the famous "Samai Bayati il-3ariyan"