By Unicode principles, a combining mark must be appear (in data stream) *after* the character it is to be attached to, so the correct notation would be `ω̑`.

Browsers often have difficulties with placing combining marks properly, though modern browsers tend to do it mostly well. Here the issue seems to be with the font. Fonts often have inadequate implementations of combining marks, especially as regards to metric information. You can see this by testing the string, ω̑  in different programs (e.g. MS Word), using different fonts. E.g., in Cambria, the rendering looks OK.

Before considering the choice of a font, or a `font-family` list, consider whether you really want to have an omega with inverted breve. Such a character does not appear in any form of Greek writing or mathematical notations that I know of. You might actually mean omega with perispomeni, ῶ, which is a normal character in polytonic Greek; it would best be written as a precomposed character, U+1FF6 GREEK SMALL LETTER OMEGA WITH PERISPOMENI, using the character reference `ῶ` if needed. You would naturally need to check that all the fonts in your `font-family` list contain it, but this is a manageable problem.