(EDIT: Your Fotron, according to
this page, apparently doesn't have any PCIe connectors, so you can't use it with the GTX 1060 anyway).
(EDIT 2:
Another site , however, says it has a single 6-pin PCIe connector)
Maybe with a HIGH QUALITY brand/model, the 350W could probably do the job. I believe a good quality 450W would have no trouble with it. The 2017 Corsair CX450 should be just fine for your system. Under full load, I don't think your system will pull much more than 300W.
Your existing PSU, if this is the label:
while it's got (barely) enough on the 12V rail, given that it's not a high quality unit (I'm deferring to
@richiestang_78 's judgment - I know FSP makes some great units and some terrible units, but I don't know which is which), I wouldn't take the chance pushing it that close. The only thing that makes me think it might be decent is that it can provide very close to its total rated power on the 12V rails.
Now, at the moment, NewEgg has a $20 mail in rebate on the CX450:
https://www.newegg.com/corsair-cx-series-cx450-450w/p/N82E16817139201
$32.98 after shipping and after the rebate is really tough to beat!
But, for only $17.01 more (again, after $20 mail in rebate), you can get yourself a good amount of extra headroom for a possible future upgrade, getting the 650W version instead of the 450W version:
https://www.newegg.com/corsair-cx-series-cx650-650w/p/N82E16817139200
The CX is the "low end" Corsairs, but arestill considered pretty good quality. 5 year warranty.
Note that the 450 only has a single PCIe 6+2 connector, whereas the 650 has two of them. I think the GTX 1060 either requires a single 6-pin, or a single 8-pin (thus the 6+2 connector), depending on which exact model of GTX 1060 you're looking at. I couldn't find if the 350W Fotron had a PCIe connector or not.