Unfortunately at the moment while the phone call option appears in the action items for contacts, there is no equivalent text button

To start a text or refer to a conversation instead you’d have to look for it in two areas: 

The phone integration: 
The phone icon will open our integration dialer and text messages can be found in here to start or read existing conversations - the integration indexes the contacts so you can just search by name or number :

Or you can add a “Text Messaging” Card to the sidebar of the contact page: 
 

If its not there - you might need to go into the record customization → click on the corresponding layout → and drag the “Text Messaging” card all the way to the top to prioritize it.