Is there a way to extend a trait in PHP?

I want to use functionality of an existing trait and create my own trait on top of it only to later apply it on classes.

I want to extend Laravel SoftDeletes trait to make SaveWithHistory function, so it will create a copy of a record as a deleted record. I also want to extend it with record_made_by_user_id field.

Yes, there is. You just have to define new trait like this:

trait MySoftDeletes 
{
    use SoftDeletes {
        SoftDeletes::saveWithHistory as parentSaveWithHistory;
    }

    public function saveWithHistory() {
        $this->parentSaveWithHistory();

        //your implementation
    }
}

I have different approach. ParentSaveWithHistory is still applicable method in this trait so at least should be defined as private.

trait MySoftDeletes
{
    use SoftDeletes {
        saveWithHistory as private parentSaveWithHistory; 
    }

    public function saveWithHistory()
    {
        $this->parentSaveWithHistory();
    }
}

Consider also 'overriding' methods in traits:

use SoftDeletes, MySoftDeletes {
    MySoftDeletes::saveWithHistory insteadof SoftDeletes;
}

This code uses method saveWithHistory from MySoftDeletes, even if it exists in SoftDeletes.