We know that an American calling himself Hargrave came to Tunbridge Wells two days ago with bicycle and valise.
In the latter was a sawed-off shotgun; so he came with the deliberate purpose of crime.

Yesterday morning he set off for this place on his bicycle, with his gun concealed in his overcoat.

No one saw him arrive, so far as we can learn; but he need not pass through the village to reach the park gates, and there are many cyclists upon the road.
Presumably he at once concealed his cycle among the laurels where it was found, and possibly lurked there himself, with his eye on the house, waiting for Mr.Douglas to come out.

The shotgun is a strange weapon to use inside a house; but he had intended to use it outside, and there it has very obvious advantages, as it would be impossible to miss with it, and the sound of shots is so common in an English sporting neighbourhood that no particular notice would be taken." "That is all very clear," said Holmes.
"Well, Mr.Douglas did not appear.

What was he to do next?
He left his bicycle and approached the house in the twilight.