"Over 45 years ago, the Ji do Kwan rose from post-World War II Korea. Members of the Ji Do Kwan supported both the creation of Tae Soo Do in 1955 and Tae Kwon Do at approximately the same time. Several well-known kwan founders trained at the early Ji Do Kwan including Hwang Kee and Won Kuk Lee. The original Ji Do Kwan was different from the other kwans. The Ji Do Kwan was used as a training hall for other arts such as Yudo and Kum Do.
In 1931, Kyung Suk Lee taught Judo in Seoul, South Korea. After World War II ended, Kyung Suk Lee asked Sang Sup Chun to Teach Kong Soo Do at the same location. Sang Sup Chun taught Dong Soo Do and then brought Byung In Lee to teach Kwon Bup, Byung In Lee then taught Yun Moo Kwan Bup Bu. Byung In Lee left the Chosan Yun Moo Kwan and taught Kwan Pub Bu at the Seoul YMCA. When Byung In Lee left the Yun Moo Kwan. Sang Sup Chun took over and continued teaching Kwon Bup until the end of the Korean War 1953.
The Ji Do Kwan or "institute of knowledge, the right way or the way of wisdom," was established on March 3, 1947, first as the Chosan Yen Moo Kwan. The Ji Do Kwan was originally named the Chosan Yen Moo Kwan and Judo was the first style taught there. The creator of the Chosan Yen Moo Kwan was Sang Sup Chun, who had studied Judo in high school and Karate as a young man. San Sup Chan fist taught Judo at the Chosan Yun Moo Kwan under Kyung Suk Lee, initially teaching Kwon Bup Bu at the facility. The Chosen Yun Moo Kwan was located first in Soo Song Dong, Seoul. After the surrender of the Japanese in June 1945, the Chosan Yun Moo Kwan Moved to the So Gong Dong, Seoul, at the Japanese Gang Duk Kwan."
http://www.dallastkd.com/jidokwan.htm
OK, I'll make a bit of a leap here - there are two things that could have happened here...(only my guess...) if Yudo was the first art taught there, they probably wore Judo style dobok, which usually have a cross-stich pattern in them - it is possible this pattern was dyed in or just got dirty, and showed - thus the cross hatching look was born.
Now, is there a direct connection between Jido Kwan and Hapkido, no, but if this eclectic approach to training was the norm for them, maybe when Hapkido came to Seoul it was adopted in the training regimine of these folks, and the "unique" uniform design stuck with others just training in Hapkido.
Just a thought.