If you're arrested by the police, they will not generally need to obtain a warrant to search your person, your pockets, your bag, or the passenger compartment of your car. Thoroughly.
People confuse "search incident to arrest" with "Terry stops", which are light, supposedly unintrusive searches that can be conducted without individualized suspicion for purposes of officer safety (ie, for weapons). Terry stops are the ones where, supposedly, the police aren't supposed to be reaching into your pocket. Search incident to arrest, going back over 100 years, has always had a purpose of discovering and preserving evidence; they aren't simply about officer safety.
That said: a modern smart phone is more than just a container you carry on your person. It can provide access to much of the same personal information that a search of your most personal possessions would. It's a totally reasonable question as to whether an intrusive search of the data on a smartphone falls into the rubrick of what search incident to arrest was meant to cover.
Either way: make sure your phone is encrypted.