How do I ping my iPhone after the new Apple Watch update?

Use Control Center on Apple Watch describes how to access the Control Center so you can "ping" your iPhone.

Open or close Control Center

• Open Control Center: Press the side button once.
• Close Control Center: With Control Center open, turn your wrist away from you or press the side button again.

Press the side button

• Press to show or hide Control Center.
• Press and hold to use Emergency SOS.
• Double-click to use Apple Pay.
• Press and hold to turn your watch on or off. 

If you have Apple Watch Ultra or later, you can customize the Action button to access your favorite features quickly.

DrGunter wrote:

My Apple Watch 2 doesn’t seem to work that way with the new update. Pressing the side button just pulls up all the open apps, allowing me to close them.

Are you sure you have watchOS 10 installed?

Settings -> General -> About

Will give you your current version.

watchOS 9 and older would bring up the recently used apps, and let you dismiss them.

watchOS 10 brings up the Control Center when doing a quick press and release of the flat oval side button

Use Control Center on Apple Watch - Apple Support

DO NOT hold the button, as that brings up Emergency services and the power off icon.

Getting the press and release timing just right can be an issue. You can make it more reliable by changing:

Settings -> Accessibility -> Side Button Click Speed

“Side Button Click Speed” can adjust the press and release timing, so you get the Control Center display every time. Adjusting The “Side Button Click Speed” is also useful for Apple Pay’s double press and release. There is “Default” (sub second timing of needed), “Slow” (up to a second between press and release), and “Slowest” (up to 2 seconds between press and release). I like the “Slow” setting.

NOTE: Slow and Slowest settings do not mean you have to wait a full second or 2 seconds between press and release. It just means you do not need to be super fast.

My Apple Watch 2 doesn’t seem to work that way with the new update. Pressing the side button just pulls up all the open apps, allowing me to close them.