There are a few options in the area of neckwear. Neckties usually seem to be narrow and tied four-in-hand (The Windsor Knot did not come into play until the 1930's). They are also worn noticeably shorter than you would today. You could also wear an Ascot tie which would have been common for day dress. Bow ties were also common. Remember to also find a matching handkerchief to put in your coat pocket to have that final polished look.
Now the upper class was still wearing top hats for formal wear, but most men would have worn a bowler or a soft felt Homburg, Trilby, or a Pork Pie if you wanted a felt brimmed hat on a bicycle. Boater straw hats and Panama hats were great summer head wear. On the soft cap side there are many choices; the Newsboy/Gatsby style was very popular, as was my favorite the ivy cap. There are also the similar, yet to a hat aficionado very different ascot and pub hat... and don't forget cycling caps!
The same neck wear was worn, but ties slowly grew broader. This is also the era when the "modern" tri-fold tie, that we all know and love, was introduced.
Hats were still common in all classes. In addition to homburg and trilby, this is the beginning of the fedora era. Regrettably the bowler was stylistically on its way out, but seems to still be pretty common in some circles. You still see the straw boater hat, especially in the summer and in the youth culture. Soft caps of the same style as discussed above were still being worn.
This was also a time of transition in timepieces. The turn of the century saw the rise of quality, affordable pocket watches. Yet the War to End All Wars changed the style of watches as it changed so many things. Because soldiers in the trenches needed to keep both hands on their rifles, they were issued wrist watches. Now this was not a new concept, but the practicality of the design started to shine through after the military broke the tradition of the pocket watch. So in the post war era it was modern and trendy to sport a wrist watch. Saying that, the traditionalist kept the stately old pocket watch well into the 1950’s.