Set the Instant Pot to saute mode and add the olive oil. Add the sliced andouille sausage and cook for 3 to 4 minutes until lightly browned. Remove and set aside.
Add the chicken thigh pieces to the pot and cook for 3 minutes, stirring occasionally, until lightly seared on the outside. Remove and set aside with the sausage.
Add the diced onion, green bell pepper, and celery to the pot. Cook for 3 minutes until softened. Add the garlic and cook for 30 seconds until fragrant.
Stir in the cajun seasoning, smoked paprika, dried thyme, salt, and cayenne pepper, cooking for another 30 seconds.
Pour in the diced tomatoes with their juices and the chicken broth. Stir to combine and scrape the bottom of the pot. Add the rinsed rice, then return the sausage and chicken to the pot. Stir gently so the rice is submerged in the liquid.
Lock the lid and set the valve to sealing. Pressure cook on high for 8 minutes.
When the cook time is complete, do a quick pressure release by carefully switching the valve to venting. Remove the lid, add the shrimp, and stir them into the hot jambalaya.
Place the lid back on (without locking) and let the residual heat cook the shrimp for 5 minutes until they are pink and cooked through.
Fluff the jambalaya with a fork, taste and adjust seasoning as needed, then garnish with sliced green onions and serve with hot sauce on the side.