Hey.

No, You cannot get admission in LLB course through AILET.

Reason being that AILET only offers 5 year integrated law course like BA LLB And not a 3 year LLB course. There are 2 types of Law courses in India- 3 year LLB course and 5 year integrated law course. 3 year law courses are for those who have completed their Undergraduation. So after completion of any 3 years of Undergraduate  course, you can do LLB which will be for 3 years.  And in 5 year law course are for those who are seeking to puruse law after class 12th. It combines an Undergraduate degree with law, for example- BA LLb, BCOM LLb, BBA LLb etc. It takes 5 years after 12th class.

As you have completed your graduation you can't pursue a 5 year Law course.

There are various other colleges that offer the course of 3 year LLB after graduation. Here are a few-

DU LLb- For admission in all centres of Delhi University 

SLAT-  For admission in Symbiosis Law School, Pune and other campuses.

LSAT- All Private Law colleges in India like IIT Kharagpur, Jindal Global Law School, UPES etc

 PU LLb- For admission in Panjab University, Chandigarh and its regional centres.

MHCET- GLC Mumbai, ILS Pune, and other law colleges in Maharashtra.  

  TS LAWCET For admission in law colleges of Tamil Nadu.

And many more

Best of luck.