If your conditional license is a learner permit, junior learner permit or junior license you may drive outside of New York State if allowable by the other state laws. Connecticut allows one to drive with an out of state learner's permit if they are at least 16 years old and adhere to the same restrictions as their home state placed on their permit. The person must also be under the instruction of person who has held a driver's license for at least 4 years.
If you instead have a restricted license or conditional driving privilege you are legally allowed to drive with certain limitations. With these restrictions you are only valid to drive to and from: work, school, DDP classes as required, to transport your child, to probation activities, certified medical exams, the DMV regarding your license and during the three hour weekly period listed on your conditional license.
If any of these activities are going to take you out of state, say to CT, you should clear it first with the NY DMV to make sure you are not breaking any of the conditions of your limited license. New York car insurance is available now for your needs.