Aye, the Snizort is probably your best bet, as Peter Ross suggests.
You could also try Loch Duagrich which feeds it. Brownies and the occasional salmon. I guess it depends on how good the runs there are these days? Drive up the B885 out of Portree, use an O.S. map to judge a good place to park, then it's about 20 minutes downhill to the loch. Watch out for a particularly boggy bit, though - my mate went up to his neck in it! Oh how we laughed.
I used to visit Skye often up to 10 years ago, so I'm rather out of touch now.
However, if there's rain then have a look at the River Hinnisdal a wee bit past Eyre on the A87 to Uig.
I used to fish here and it's an interesting little spate stream. Walk the mile or so down to the sea loch even if there's no water in the river. Fish often used to nose their way up the estuary with the tide. I saw grilse there as well as sea trout and finnock, but only ever hooked the latter two.
It's a lovely place with a few trees along the banks, old ruins nearby and even a sheltered sandy spot down at the estuary on the north bank (Uig side), perfect for a lunch break. Park just before the road bridge in the big lay-by on your right (coming from Portree).
If you like a stiff walk, try Loch Fada above Camustianavaig, near Braes (B883), about 3 miles south of Portree. It's not far from the road but it's a steep climb! Brownies only, but beautiful fish. There's a good head of shrimp in the loch which helps them along - and it's downhill all the way home!
The Skeabost Hotel used to manage the Snizort and I think Ullinish Lodge had Duagrich (and L. Connan, a nice trout loch nearby). Nobody was looking after the Hinnisdal during the years I visited. I think that the Portree Angling Assoc. used to have L. Fada (Braes, as well as Storr), but their website doesn't mention it now, so maybe ask at Jensport in town?
Ah! This site has more up-to-date info on permits:
Skye and Lochalsh Angling
All the best.