Write the three-digit number as $abc$. The number we obtain by reversing the order of the digits is $cba$. The sum $x$ of these two numbers is
$$x = \overline{abc} + \overline{cba} = (a + c)10^2 + (b + b)10 + (c + a).$$
Since all the digits of $x$ are odd and $2b$ is even, we conclude that $c + a$ is equal to at least 10. The number $c + a$ cannot be equal to 10 because the units of $x$ are odd and equal to the units of $c + a$. Hence, $c + a$ is at least 11 and $a$ is at least 2. The number $abc$ will be the smallest when we choose the smallest possible $a$, i.e. $a = 2$. In this case $c = 9$ and the smallest possible value of $b$ is

We see that the number $209 + 902 = 1111$ consists of only odd digits. Thus, $209$ is the smallest number with the desired properties.