how to increase the current from the LTC3108 chip
The problem is not with the chip (well, not the main problem anyway - but we'll get to that), the problem is with the rubbish efficiency of peliters at small temperature differences, first, the problems.

Peltiers are normally used to create temperature differences when a current is applied to them, peltiers are not known for their stellar efficiency, and when used as generators they're even worse, I've seen figures like 5% labeled as "cutting edge efficiency", so that's our first problem. So you might need to push 100W of heat through a peltier to get 5W of electrical power.

Our second problem is that peltiers have low thermal resistances, see thermal resistance is the temperature difference created by a flow of heat energy (measured in degrees per watt 'C/W). A low thermal resistance means you need a powerful heat flow (not necessarily a high temperature, but a powerful source, lots of watts) to generate any meaningful temperature difference across the device, and we need a big temperature difference across the peliter if we want to generate any meaningful amount of power. This is part of the reason why their efficiency is so low, you need a big temperature difference to get lots of power, but they conduct heat too well - they maintain a high temperature difference about as well a screen door on a submarine maintains a high pressure difference.

This leads to the third and least intuitive problem: Getting a decent temperature drop across the peltier in the first place. See, there's not a great deal of thermal energy available for harvesting from a person in the first place, maybe 200W max across their whole body (which I think is something like 2m^2) so first of all, that puts an upper limit on power generation (the most common peltier is 40x40mm or 1/2500th of a person, so that's already ~0.1W per peltier MAX). And once you take into account the pitiful generating efficiency of peltiers, suddenly you're looking at only a few milliwatts per 40x40mm device (stacking them helps a bit, but while 10mW is twice as good as 5mW, in the grand scheme of things...) This is made worse by the fact that the ability of the peltier to dump the waste heat (all 90-95% of it) into the environment, which is very strongly dependent on the temperature difference, hot things lose heat faster and more easily than mildly warm things, so you need big heatsinks to compensate.

Now you know why chips like the LTC3108 are only designed to spit out a few mA, the kind of applications that thermal harvesting often gets used in uually have so little potential generating capacity that's it'd be pointless using anything bigger. (and mainly the term "energy harvesting" is only applied to really * really* low power stuff)

Now I could be out by an order of magnitude in my calculations, but even then, 5W is 1000 times greater than what I'd expect to be able to generate from a single peltier strapped to a person. If you stuck a big ol' CPU cooler on one side and put a blowtorch on the other, then I'd expect to get 5-10W out of each peltier.

(as for a supercapacitor helping things along, it's not that having a big cap allows you to generate more power, it's just that it allows you to store the energy you've harvested and then release it all in one big burst - like sending a radio packet - of course, now that the cap's been drained, you have to wait a while for it to be recharged)