So, i see there's been some confusion as to how you unlock gunsmith part 5, and what requirements you're supposed to meet before obtaining the quest.
I've seen multiple players in different levels not having the quest after gunsmith part 5, but it seems like its time-locked at around 24 hours after completing gunsmith part 5.
I got this quest completed at 5:45AM (Picture showed time yesterday, but doesnt now unfortunately), and as of time writing 6:35AM i've unlocked the quest without completing any quests inbetween.
Im making my assumption based on what the mechanic says after completing gunsmith part 5: "Great gun, a solid build. There are no new orders, but you can come by tomorrow"
I dont want to take credit for this, but i've not really seen anyone "confirm" it, but i'd say this pretty much does.
Edit: Timestamps for completing quest
Source: https://www.reddit.com/r/EscapefromTarkov/comments/101zjeo/gunsmith_part_6_confusion_solved/