If you need to type some Greek and for some reason cannot activate or install an input system on the computer (or if typing Greek is not something you plan to do again), you can try this online “keyboard” from Lexilogos. When you click a pictured letter (including letters with diacritics), it appears in the text box. The letters highlight as you hover over them, so it’s easy to see which you’re clicking. When you’re done, just copy the text and paste it wherever you want.
You can also use your physical keyboard according to the instructions on the page. I found typing by keyboard with Lexilogos a bit hit-or-miss. On the one hand, it does some clever things like convert your typing “th” into a Greek theta (both “ph” and “f” will give you phi). On the other hand, I couldn’t get some diacritics (grave, macron, breve), but I may have misunderstood the instruction. I’m also not sure of the rationale for using an ampersand to add a circumflex, as opposed to the “pointy hat” circumflex or the tilde. Fortunately, you can alternate as much as you like between “point-and-click” entry and “keyboard” entry.
Lexilogos includes a few less common characters, like koppa and sampi, but I don’t see a lunate sigma.
This online keyboard is definitely meant to be used on a desktop/laptop computer, but it seems to work (albeit awkwardly) on iOS, so if you’re really in a pinch to get a particular character on your iOS device, it’s worth a shot.