About the stability of the tangent bundle restricted to a curve

Let C be a smooth projective curve of genus g >= 2 over an algebraically closed field k and let L be a line bundle on C generated by its global sections. The morphism phi(L):C -> P(H-0(L)) similar or equal to P-r is well-defined and phi(L)*T-Pr is the restriction to C of the tangent bundle of P-r. Sharpening a theorem by Paranjape, we show that if deg L >= 2g - c(C) then phi(L)*T-Pr is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree 2g - c(C) - 1 such that phi(L)* T-Pr is not semi-stable. Finally, we completely characterize the (semi-)stability of phi(L)* T-Pr when C is hyperelliptic.