Is a veil too religious for a secular ceremony? I don't want anything too religious

Not at all. A veil is very elegant and assosiated with a bridal look than any other accessory. 

A veil can be the perfect finishing touch for almost any wedding gown.

I have made them for beach weddings, winery weddings, church weddings, historic home ceremonies. There are so many styles to choose from a tiny blusher to a catherdral length, double layer, single layer, lace edged or plain. Have a look at some of the designs I have made for brides over the last 5 years https://www.easyweddings.com.au/WeddingAccessories/Melbourne/EmpireRoomandAlterations/Gallery/

A veil is gorgeous you should have what you feel is right for you. Don't have to be religious to wear one. How about a bridal comb for when you take your veil off.