Given what you currently have I would strongly suggest not doing the Sigma 10-20 f/3.5. Instead you should really consider renting or buying a telephoto lens. Something like a 135mm or 200mm of really any aperture will work since you'll be using this on a tripod most of the time. Getting reach for framing the shot is very important in landscape. A 70-200 f/4 is also an option that a lot of people go with for this very reason.
Then if you're tripod isn't good you might want to buy a travel tripod. Hard to say exactly since you didn't mention the exact one you have. I love my discontinued Benro Travelflat because it goes entirely flat. As mentioned its discontinued so would have to check ebay/keh type places to find it, or check out Benro's other offerings. Another really popular travel option is BeFree by Manfrotto (I almost bought this until I saw the shop had the Benro in as a used item for a sizable discount).
Typically landscape photographers use a Ball Head but it can be done with any head. Architecture photography at the high level uses a geared head but would also be using a Tilt Shift lens. Based on your question I don't think this is you so I wouldn't worry about either a TS lens or a geared head.
Other good things to have from a strict photography sense depending on budget would be ND filter, polarizing filter, remote release.
I would however only recommend those if budget allows after the lens and tripod AND after safety and hiking gear has been bought. Rain protection, sun protection, comfortable bag, etc.
Here's some other questions you might want to check out: