There are three ways to carry an “orthodox” shilellagh, and they refer to how you recover the shillelagh after each walking step.
Palm carry is where you grip the head from above, with the bulb sitting in your palm, and friction holding the shillelagh in place as you walk. It’s immersive, traditional, and suits small-headed shillelagh well.
Ring carry is where you wrap your thumb and index finger into a ring around the head of the shillelagh, you can comfortably carry any shillelagh this way, and is especially comfortable if you tend to wear longer staves (ie where the head is at navel, rather than hip level).
Cane carry is good for New England or “Claymore” style shillelagh where enough of the head comes forward over the index finger. This style shillelagh can be recovered using just the index finger. It’s the least traditional , but arguably the most comfortable, and paradoxically most immersive way to recover a shillelagh.
Ultimately its up to what you want from your shillelagh and what you want from your walk that day. On a day when the troubles of the world are weighting on you, take a heavy, palm-carry staff on your walk, to draw you out of the clouds and into the countryside around you.
Strolling with your wife (or indeed someone else’s) take a light cane-carry staff that won’t distract you, and will be easy to carry if you need to head off at the run.