As you say, 가지고's original meaning is "having", or "with (smth)", coming from 가지다. 연필 가지고 장난하는 아이: a child playing with pencils. 서류를 모아 가지고 은행에 갔다: collected the papers and took them to the bank. This sense of "with smth" is then extended to action and state, becoming "do smth, then with the result/state of it...". It is like connecting the two actions with "and" but with emphasis on "making use of the result of the action". 준비를 다 해 가지고 떠났다: we did all the preparation, then left X에 Y를 더해 가지고 2를 곱해: add Y to X, then multiply the result by 2 엘에이에 가 가지고 맛있는 거 많이 먹고: go to LA and have lots of delicious stuff.. 이래 가지고(서)는 = 이렇게 해 가지고(서)는: things going this way; with this state of things 저개 가지곤 = 저렇게 해 가지고는: with things going that way; seeing that... This sense is still further extended to "on account of something", "owing to something", or "because". 어려운 수학 문제를 가지고 씨름하다: grapple with a tough math problem. 그 문제 가지고 너무 걱정하지 마: don't worry too much about that issue. 친구의 일로 고민하다: trouble oneself on account of a friend's matter 앞으로 너무 가 가지고 뒤로 빼야 했다: because we overshot to the front we had to back up So we can say 가지고 matches English "with..." in a broad swath of its meanings.

When the verb 가지다 is used as "aux verb", it shows that the activity or action of main verb is completed/finished, and the result of activity keeps on going since then. This is very similar to the completion use of perfect aspect of verbs as in English. And it is always used in the verb phrase form of "main verb stem + '-아/-어/-여' 가지-고" especially in a spoken speech. In fact, the function of aux verb '가지고' is exactly the same to the connective ending '-아서/-어서/-여서' in the meaning. For examples, 어젯밤에 우리 아빠가 크레파스를 사 가지고 오셨어요. Last night my father bought crayons and come back home. 사 가지고 오셨어요 or 사(서) 오셨어요. LA에 가 가지고 (the verb 가다 is the main part.) or LA에 가서 (using connective ending '-아서') 너무 아파 가지고 눈물이 났어요. (the adjective 아프다 is the main part.) or 너무 아파서 눈물이 났어요. (using connective ending '-아서')