You can use 'Ultimate Thread Group' instead of regular Thread Group. There is 'Initial Delay' in ultimate thread group which will help you to achieve your requirement.
stepping thread group
In the 'thread scheduling parameters' section, you'll note you're able to specify the maximum number of threads to run at peak, but you're also also to slowly instantiate threads as you need them.
As per your Test Plan description, you can use 1 of the timers mentioned below:
Constant Timer would be best option as per your mentioned requirement.
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