jmc

Of the original five positive integers, we know that the median value, or the medium-valued number, is 4. Since the unique mode is 3, there must be at least 2 threes, and since this is less than the median, we know both of the smallest numbers are 3. Finally, since the mean is 4.4, the sum of the five integers is $5\cdot 4.4=22$, and $22-3-3-4=12$ is the sum of the two largest integers. And since the mode of 3 was unique, we know that the larger digits must be greater than 4, and cannot both be 6. So they must be 5 and 7. So our set of integers is $\{3,3,4,5,7\}$. By adding in an 8, we move the median to $\frac{1}{2}(4+5)=\boxed{4.5}$.