About the stability of the tangent bundle restricted to a curve

Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and let L be a line bundle on C generated by its global sections. The morphism i:C -->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent bundle T of the projective space P. Sharpening a theorem by Paranjape, we show that if deg L>2g-c(C)-1 then i*T 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 i*T is not semi-stable. Finally, we completely characterize the (semi-)stability of i*T when C is hyperelliptic.