Let's take the scale in question and spell it out in A min: A B C D E F A. Just as you've said, if you remove the fourth (the D), you have a Hirajoshi scale in A. As your question states, your motivation for finding a repeating intervalic pattern is based on the Arabic scale, which is marked by this intervalic pattern: 1 semitone - 3 semitones - 1 semitone. In C, this Arabic scale gives:
The 1-3-1 semitone pattern is seen starting from C (C Db E F) and again from G (G Gb B C).
By contrast, here is the repeating intervalic patten in the unmodified Hirajoshi scale (in semitones): 4-1. This can be characterized as the distinguishing feature of the Hirajoshi scale, which contrasts from the 1-3-1 distinguishing feature of the Arabic scale. In the key of A, this 1-4 pattern is seen two places in the Hirajoshi scale: we see it starting on B (B C E) and also on the E (E F A). So using / stacking that 1-4 interval pattern twice produces the Hirajoshi scale. As we would expect, the Hirajoshi scale has fewer intervals in its repeating pattern because the Hirajoshi scale has fewer notes than the Arabic--the Hirajoshi scale is a pentatonic scale (5 notes), whereas the Arabic scale is a heptatonic scale (7 notes).
When you take the Hirajoshi scale and add the extra tone (an F, when dealing with the key of A), the intervalic pattern I'm hearing (in semitones) is: 2-1. This pattern is seen beginning on A (A B C) and D (D E F). Combining those six notes produces the modified Hirajoshi scale you've described.
So in both cases--the pentatonic Hirajoshi and the modified Hirajoshi--your intuition is correct. There is a repeating intervalic pattern that, when stacked twice, forms each scale.
Given all of this, I don't think adding notes to the Hirajoshi scale would emphasize the particular genre of traditional Japanese music that you're picking up on. The characteristic interval is 1-4, and the 4 is an especially large jump (a make 3rd). As a result, there's simply no room left within an octave for additional stacking of this particular pattern.
Note: there's a second reason why extra stacking or extra notes would not emphasize the scale's traditional Japanese sound. The Hirajoshi scale is a pentatonic scale. Songs and solos written around a pentatonic scale often invoke creative and unique-sounding intervals in the melody line, and in think this is part of the characteristic sound you're describing. To illustrate this in an exaggerated example, consider this ascending scale pattern from a Hirajoshi scale in A: A4 C5 E5 B5 C5 F5 A5 E5 F5 B6 C6 A5... This pattern is formed from broken "thirds" (1-3, 2-4, 3-5, etc.) which alternate ascending and descending (1-3, 4-2, 3-5, 1-4.) When we add the extra tone into the Hirajoshi scale (the F when working in the key of A), we remove some of the larger intervals between scale tones, which can remove the more creative intervals in melody lines. This pentatonic nature is a crucial part of the Hirajoshi sound, and adding extra notes would remove this potentially defining quality of the genre from the scale.