Let
$K$
denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of
$K$
is said to have property
$T\left( k \right)$
if for every subset of at most
$k$
translates there exists a common line transversal intersecting all of them. The integer
$k$
is the stabbing level of the family. Two translates
${{K}_{i}}\,=\,K\,+\,{{c}_{i}}$
and
${{K}_{j}}\,=\,K\,+\,{{c}_{j}}$
are said to be
$\sigma$
-disjoint if
$\sigma K\,+\,{{c}_{i}}$
and
$\sigma K\,+\,{{c}_{j}}$
are disjoint. A recent Helly-type result claims that for every
$\sigma \,>\,0$
there exists an integer
$k\left( \sigma \right)$
such that if a family of
$\sigma$
-disjoint unit diameter discs has property
$T\left( k \right)|k\ge k\left( \sigma \right)$
, then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval
$k$
. The asymptotic behavior of
$k\left( \sigma \right)$
for
$\sigma \,\to \,0$
is considered as well.
Katchalski and Lewis proved the existence of a constant
$r$
such that for every pairwise disjoint family of translates of an oval
$K$
with property
$T\left( 3 \right)$
a straight line can be found meeting all but at most
$r$
members of the family. In the second part of the paper
$\sigma$
-disjoint families of translates of
$K$
are considered and the relation of
$\sigma$
and the residue
$r$
is investigated. The asymptotic behavior of
$r\left( \sigma \right)$
for
$\sigma \,\to \,0$
is also discussed.