No you don't need a spanner tool, any seat adjustment should be done before you leave, and besides you shouldn't have to adjust your seat spanner more than once every couple of years after it's broken in.
What tools to take along for a ride depends on how good you are at repairing stuff, and or how far you ride from home. I carry a Park MTB 3.2 multi tool because I can do most repairs myself and this tool will do most repairs. I also carry a cheap pair of folding pliers.
Even if you don't know how to work on your bike you at least need to know how to fix a flat, so you need a pair of tire irons, I like the Soma Steel Core Levers because they're the toughest made and won't mar your rims. On my touring bike I also carry a VAR tire lever (in addition to the tire levers), this tool will get on the last inch of even the toughest tire to install without chance of damaging the tube. My touring bike has very difficult to install tires on it so this tool is very handy.
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Another great tool is the QuikStik, this makes removing and installing a tire fast, once the levers get the tire started simply insert the QuikStik and rip the tire off; installing a tire you just reverse the action and it will get right up to the last inch or so where a lever will finish it. Of course some tires can be put on entirely by hand, this should always be attempted first before using a tool.
Obviously if you'll be carrying tire tools then you need a pump, but be careful of which pump you buy because about 98% of the pumps on the market will not get you to 100 psi or so, in fact most fail at about 75 regardless of their claim that they can reach 160 which I haven't found any carry on bike pump that can go that high, not even the much better full size frame pumps. The only mini pumps that I've found that can get your pressure up to 100 to 120 psi (note I said what I found, there maybe a couple others that can) are: Lezyne Road Drive large (they offer 3 sizes the smaller ones will not, the middle size might but it would be a lot of strokes and a lot of physical strength); Topeak Road Morph G (this pumps suffers from a poorly designed mounting system, and ungainly appearance), this pump is the easiest of all mini pumps on the market but it's more of a half frame pump than a true mini pump, this pump takes about 175 strokes to get to 110 and with the fold out foot peg and T handle it does it like a mini floor pump...very easy; Topeak Race Rocket HP, this one will take about 75 more strokes to get to 110 than the Lezyne Road Drive large but it will get there. I own all those pumps, I owned others but they failed completely to deliver the PSI I needed so sent them back for refunds or exchanges. I also own a SKS Wese RaceDay Carbon pump, this one does get to 110, it takes more strokes than the others I have but it's also the lightest of the others, sadly they no longer make this pump and SKS now makes mini's that won't do the job. I have a friend who has the Zefal Air Profil LL Mini and that pump can also reach 110 but it takes more strokes than the Lezyne Road Drive large, but less than the Topeak RaceRocket HP.
Also carry a patch kit and learn how to patch on the road, a spare tube is essential but if by chance you flat the spare you better know how to patchs. I use only glueless patches and have done so for over 20 years with only 2 failures. How to use these correctly is the key for the patch to last the life of the tube, otherwise glue on works just fine and covers any mistakes you might make in preparing the tube for the patch.