  In this paper, we study the relationship between the radius $r$ and the
attachment number $a$ of a tetravalent graph admitting a half-arc-transitive
group of automorphisms. These two parameters were first introduced in~[{\em
J.~Combin.~Theory Ser.~B} {73} (1998), 41--76], where among other things it was
proved that $a$ always divides $2r$. Intrigued by the empirical data from the
census~[{\em Ars Math.\ Contemp.} {8} (2015)] of all such graphs of order up to
1000 we pose the question of whether all examples for which $a$ does not divide
$r$ are arc-transitive. We prove that the answer to this question is positive
in the case when $a$ is twice an odd number. In addition, we completely
characterize the tetravalent graphs admitting a half-arc-transitive group with
$r = 3$ and $a=2$, and prove that they arise as non-sectional split $2$-fold
covers of line graphs of $2$-arc-transitive cubic graphs.
