Quasi-categorical variables - any ideas?
I would suggest you to use the idea of so-called 'fuzzy clustering', where you put each of your hours of the day value into several clusters at the same time. Details in paper: http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/cmeans.html

The idea is trivial:

You decide how many clusters you want to have. For example, 4 (so you divide your day hours into 4 cathegories). Instead of computing just 1 number (which defines cluster membership) for each of your day hours you compute 4 numbers which represent the degree of membership to each of 4 clusters. So for example if you 4 clusters will contain periods 12 AM-6 AM, 6 AM- 12 PM, 12 PM - 6 PM and 6 PM - 12 AM then you would replace for example 4 AM hour in original data with vector of 4 numbers, first one is the biggest, second is smaller, third one is the smallest one etc.

Then you could use these 4 numbers in your model to fit a regression line.

Of course, if you want you could use 24 clusters and in such case each your day of hour would have a high 'relation' with nearby hours and almost 0 with the distant hours.