At first we looked at Laugavegur, but after reading more about it, it doesn't sound very wildlife-like
What kind of wild-life were you thinking of seeing on a hike in Iceland in September? Well, none, so let's put that reason for not doing a particular hike to one side. So let's remember that Laugavegur is the most famous hike in Iceland because it has the greatest variety and interest of scenery of any hike in the country, is relatively civilised in terms of waymarking and visible path underfoot, and work out more sensible reasons why you might or might not be doing it in mid-September.
Although the last bus to Landmannalaugur is scheduled on 18 Sept
http://www.nat.is/re/bus_stop_rek_landml_rek.%20re-htm.htm
it doesn't always succeed in running that late, because it can be snowed in before then, I think the 9th was the earliest in recent years. So you will arrive there needing to check whether it is still running. Then you shouldn't set of on the hike at this time after the hiking season unless you have assured yourself that there is nothing in the weather forecast for the next few days that suggests any possibility of you getting snowed in while you are on the hike. Because if that happens, it suddenly turns from a hike into a winter mountaineering expedition, and it will be a battle between hypothermia and helicopter. There have been recent years when you would have been fine doing it in mid-September, but probably the odds are rather worse than 50%.
So, you can gamble, but you'd have to decide at the last minute whether it was on or off, and have an alternative possibility in mind.
Most of the famous hikes in Iceland have similar issues, so I have difficulty suggesting alternatives you can reliably do. Probably the safest option would be the Dettifoss-Asbyrgi hike, but that's only a 2-day hike. Someone else just posted asking about Hornstrandir, but the boats stop running at the end of August.
You can research the network of paths called Viknaslodir in the east of Iceland, there's an overview map and description of the options on www.nat.is. You can probably walk in Snaefellsjokull Nat Park, but it isn't very exciting.
