Establishing trust when entering a new team often translates to building a rapport and keeping your head down while learning the ins and outs of your new environment before suggesting any improvements. This equates to working side by side with those who are more familiar with the environment, while seeking their insights, asking questions, getting a clear understanding of their systems while allowing them to get to know you. This is where the 30/60/90 rule comes into play.
The first month of a new hire's 90 day probation period should focus on familiarizing themselves with the team, company, products/services, clients, etc. Month two should be spent studying current processes and procedures, identifying strengths and weaknesses. In the last month, that hire should be well-grounded in the role and capable of identifying growth opportunities. Past the 90 day mark is considered an acceptable time to implement innovation.