Separate, Different, Another, Separate Thing, Different Thing, Another Thing
の adjective
The の is there to indicate that it is the difference of something. For example this is like saying "the separate ______" or "the different _____" or "another ______". 別 by itself just means separate, different, or another.
べつの
The reading is the same as the one you learned. Move on to a 別の漢字, ok?
別のアプローチをしてみない？
Why don't we try a different approach?
昨日デートに行ったのは別の男です。
It was another man who I went on a date with yesterday.
姉は私とは別の母から生まれた子です。
My sister was born from another mother.