Notice that the word changes once we leave the Shire?  In the OED entry, the hyphenated form is associated with older quotations, the unhyphenated form is from the 1900 or later entries.

• 01.097 As soon as I saw your funny faces on the door-step,
• 01.140 If you sit on the door-step long enough,
• 01.141 and isn’t sitting on the door-step your job,
• 05.148 his nice brass buttons on the doorstep.
• 08.127 Buttons all over the doorstep!
• 11.020 Bilbo found sitting on the doorstep lonesome
• 11.020 there was not a doorstep,
• 11.020 and the opening the ‘doorstep’
• 11.020 when he said they could sit on the doorstep
• 11.022 You said sitting on the doorstep
• 12.081 in a faint on the ‘doorstep’.
• 13.052 But I don’t feel that Smaug’s front doorstep

“ˈdoor-step, n.” OED Online. Oxford University Press, June 2017. Web. 7 September 2017.