The main limitation on the number of actions is the amount of time in the day. You could simply cut back on sleep and accept being tired most of the time for most things. This will hamper at least some of your actions, and make some things harder, but you should be able to cut back enough to get an extra action each week.
You could also cut sleeping even more, along with some other actions such as eating decent meals and keeping yourself properly clean. This would give two extra actions a week, but you'd be exhausted, not merely tired, and there would undoubtably be other issues related to smelling badly and wearing dirty clothing, not to mention potential health issues.
You could reduce some of the issues with tiredness by drinking coffee, but that's expensive, and would cost about a silver a week.
As for magic or more extreme possibilities, you aren't aware of any commonly used techniques for such, but it'd be hard to say there wouldn't be any possibilities, just that there are none in common use.
For magical theory and practice, there are a few options. You could do self-study. There aren't any huge free libraries, but you know there are always lots of students and prospective students studying at the main academies that people wouldn't look too hard at there being one more.. at worst, you might get hassled a bit or made to leave, and at worse you miight need to pay a small bribe or two to make sure you aren't noticed. And you might be able to sit in on lectures and demonstrations and such if you're sufficiently agressive and lucky. This could also be aided by purchasing other study materials or textbooks for personal use.
You could also hire a private tutor. This would be significantly expensive, maybe between two and four silver a session, but you'd get much more effective instruction. It would also be dependant on the quality and abilities of the tutor in question, but you would have quite a bit of choice as to who you hire to teach, as well as the day-to-day curriculum.
You could sign up as an apprentice in some field as well. You'd be expected to pay the apprenticing fees of about twenty silver, but you'd get some chances to learn some of the practice. For this field, an apprenticeship is unlikely to lead to a proper formal knowledge of what it is and how it works, but a, you should learn some things, and b, you would be getting paid. It would also leave you with not that much choice as to who you get apprenticed to.
You could enroll formally with an institution. You aren't certain how much this would cost, although you think it'd be comparable to private tutoring. To it's advantage, you'd have much more access to self-study materials if you want to spend extra time learning. However, you would have much less control as to the quality of the instructors and the content of the course materials.
Lastly, if you have just a few questions, you could just find a mage and attempt to get information from said mage. Generally people are a bit reluctant to tell trade and guild secrets, but magic is complicated, and it usually isn't that hard to get a few questions answered. This would be much more effective if you had a favor owed to you, or something to trade.
Do note that these aren't exhaustive possibilities. If you have an idea different than the above, feel free to go with it.
Week 8
Assets and Abilities:
Prestige = 2
25 silver.
Concealable fighting dagger
Shun Ziji's dark blade.
Steel Dao. +1 Combat.
Assassin's signet ring.
Adventurer's diary (partially studied)
->Information on southern jungle nations
->Secret locations in southern jungle.
->Information on spiritual beings, spiritual being society, spiritual being outlook.
->Information on animal life of the southern jungle
->Information on magical beings of the southern jungle
->Instruction on spirit-related magical rituals. Versatile dust triggering rune. Thought communication? rune? One ritual remains.
->Instruction on an unknown magical field.
->Instruction on wilderness survival methods
->Instruction on what seems to be the language of the southern jungle nations.
->Instruction on an unknown language.
Miscellaneous magical diagrams, captured from one of your dreams. Uncertain accuracy.
Skills:
Charming, +1
Combat, +1/+1. Unarmed combat training. Also applies while armed.
Magic:
Versatile Dust triggering rune.
Allies:
Huan Zhang, diplomat, 2 prestige, 0 favors.
Li Song, martial arts student, 0 prestige, 0 favors.
Upscale Weapon Shop.
Enrolled in advanced Dao training, -2 actions a week, -2 silver.
Appointment for week 9 to have the Black Blade fully identified.