Not necessarily. You really don't need a core text other than the Children's Bible and a selection of the great children's books recommended for the Beginner level. I do encourage you to use the Companion Reader, even if you don't read it to your child, because it includes important historical information that will help you to understand and explain the historical events covered in each unit. Remember, this is a family-based program. Your children aren't the only ones who are learning!
The goal at that age isn't to learn every detail of history, but to get an overall understanding of the flow of salvation history and lay the foundation for later, more detailed, learning. This is a time to encourage interest and enjoyment. Children tend to naturally enjoy history, especially when it includes fun books and hands-on activities.
Remember that Connecting with History is a cyclical program. Always keep in mind that your first grader will come back to this Volume when they're older. That will be the time to go into more historical depth. Right now you're laying the foundation for that later study.
It's actually quite recent that I even included a textbook in Volume One. I consider the children's Bible the "core textbook" with the other core books oriented to the Bible readings. That helps to emphasize the connections between the people and events from the Bible and their interaction with the surrounding cultures. They hear the familiar Bible stories in a new way - not just "stories" but a part of a larger history of peoples and times in world history. The whole program is about making those kinds of connections (Connecting with History!)
Some people do like to have a core textbook on hand to either read aloud or for their personal reference as they move through the year. This is fine if you think it will be helpful to you. Also, if you're teaching a Grammar level student as well it's a great idea to read their Core text aloud so that everyone can listen in.
Another resource to provide cohesiveness for a young child is to use the Rhyme-line Cards and Sing-Along Songs. Memorize them in chronological order to familiarize your child with the major people and events of the time you're studying. This will provide an introduction to future studies.
Make sure that you include hands-on activities in your history studies, especially with young children. They will internalize their lessons through making simple crafts, acting out stories, etc.