Is there a standard construction of a metric on one-point compactification of a proper metric space?

Comments:

• A metric space is proper if all bounded closed sets are compact.
• Standard means found in literature.

A possible construction:

Here is a simplification of the construction given here (thanks to Jonas for ref). Let $d$ be the original metric. Fix a point $p$ and set $h(x)=1/(1+d(p,x))$. Then take the metric $$\hat d(x,y)=\min\{d(x,y),\\,h(x)+h(y)\},\ \ \ \ \hat d(\infty,x)=h(x)$$

• Is this a standard construction?
• Does it have a name?