Suggestion:
In the Fallback class do a check if the WebViewActivity is callable else open the url (Uri) with a standard action view intent:

        Intent i = new Intent(Intent.ACTION_VIEW);
        i.setData(uri);
        activity.startActivity(i);

In MainActivity, CustomTabActivityHelper.mayLaunchUrl can't be called during onCreate: neither the client or session are created because their creation happens after onCustomTabsServiceConnected callback, which is called in onStart.

The CustomTabsSession.mayLaunchUrl optimization needs to be called after the service connected callback (where warmup occurs).

I address this in my fix for CustomTabActivityHelper which changes mayLaunchUrl to set the Uri and Bundles as field variables and session.mayLaunchUrl is called after receiving the service connected callback. This allows you to keep MainActivity the way it is and let you set the preferred URL without waiting for the service callback within the Activity.

When I click launch custom tab, APP would crash certainly. It happens in HTC butterfly, but it work fine in Nexus 5. The log is below:

09-09 16:15:38.700 E/AndroidRuntime(14071): FATAL EXCEPTION: main
09-09 16:15:38.700 E/AndroidRuntime(14071): Process: com.hitherejoe.tabby, PID: 14071
09-09 16:15:38.700 E/AndroidRuntime(14071): java.lang.RuntimeException: Unable to start activity ComponentInfo{com.hitherejoe.tabby/com.hitherejoe.tabby.ui.activity.WebViewActivity}: java.lang.NullPointerException
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread.performLaunchActivity(ActivityThread.java:2596)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread.handleLaunchActivity(ActivityThread.java:2653)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread.access$800(ActivityThread.java:156)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread$H.handleMessage(ActivityThread.java:1355)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.os.Handler.dispatchMessage(Handler.java:102)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.os.Looper.loop(Looper.java:157)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread.main(ActivityThread.java:5872)
09-09 16:15:38.700 E/AndroidRuntime(14071): at java.lang.reflect.Method.invokeNative(Native Method)
09-09 16:15:38.700 E/AndroidRuntime(14071): at java.lang.reflect.Method.invoke(Method.java:515)
09-09 16:15:38.700 E/AndroidRuntime(14071): at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:1069)
09-09 16:15:38.700 E/AndroidRuntime(14071): at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:885)
09-09 16:15:38.700 E/AndroidRuntime(14071): at dalvik.system.NativeStart.main(Native Method)
09-09 16:15:38.700 E/AndroidRuntime(14071): Caused by: java.lang.NullPointerException
09-09 16:15:38.700 E/AndroidRuntime(14071): at com.hitherejoe.tabby.ui.activity.WebViewActivity.onCreate(WebViewActivity.java:26)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.Activity.performCreate(Activity.java:5312)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.Instrumentation.callActivityOnCreate(Instrumentation.java:1111)
09-09 16:15:38.700 E/AndroidRuntime(14071): at android.app.ActivityThread.performLaunchActivity(ActivityThread.java:2552)
09-09 16:15:38.700 E/AndroidRuntime(14071): ... 11 more

hi,i download your demo,but when i run , DaggerApplicationComponent do not exist

https://github.com/hitherejoe/Tabby/blob/master/app/src/main/java/com/hitherejoe/tabby/ui/activity/MainActivity.java#L184

and

https://github.com/hitherejoe/Tabby/blob/master/app/src/main/java/com/hitherejoe/tabby/ui/activity/MainActivity.java#L177

You should use the PendingIntent.FLAG_UPDATE_CURRENT flag.
It's not really useful in your case as you always open the same Url. But if someone is using this code for multiple websites, the PendingIntentwon't be overridden and the share / email will be performed on the first opened website.