"The old inn at Lundin Links, which has been well known in the locality for many years, if not centuries, is now in course of demolition. Like many old and deserving institutions, it had to give way to modern ideas and usages and to accommodate itself to the development of the place."
This strongly suggests that the old in was retained and kept in business until the new building was completed and ready to take over. As the new hotel took two or three years to build, the two must have co-existed for a time. I wonder if any photographs exist out there of the two side by side. Just to check the feasibility of this co-existence, I checked some old maps to see the relative positioning of the buildings. Overlaying a map from 1893 with one from 1912 shows that the two could easily both fit within the plot (see below).