姐 vs 姊, etymology, and their English equivalences
「姐」 (Zhengzhang OC: /*ʔsjaːʔ/, mother > senior female > adult female, elder sister) is comprised of semantic 「女」 (female) and phonetic 「且」 (/*sʰjaːʔ/). Through its semantic extensions, 「姐」 has developed into many uses:


The meaning big sister is roughly equivalent to English big sis', more commonly used in colloquial spoken language;
The meaning mother is no longer found in Standard Mandarin, but is preserved in some other topolects;
All other meanings are from an extension of adult female. Your reading of 「小姐」 does not imply that 「姐」 means junior female, but rather the prefix 「小」 being used as a term of endearment. Please see the question At what age is it inappropriate to call people 小李, 小王, 小刘, etc.? And is it different for males and females? for an idea of how 「小」 works.




In contrast, 「姊」 (/*ʔsiʔ/, elder sister) is comprised of semantic 「女」 and phonetic 「」*. The meaning elder sister is frozen, and is the literary vocabulary used in writing the word for elder sister.

*「」 is inferred to be similar in sound to 「次」 (/*sn̥ʰis/) through readings of oracle bone inscriptions.