Let’s break this passage down clause by clause.
A few minutes later we had reached the lodge-gates...
This fragment is easy.
A few minutes later we had reached the lodge-gates…
Time person action thing/place…
There are no fancy gothic words you need to use. Keep it simple.
A few minutes later we had arrived at the terminal…
A little while later I had reached the shore…
Some time after that they reached the top of the hill…
