We simulate the evolution of eccentric binary systems in the frame of the grazing envelope evolution ( GEE ) channel for the formation of Type IIb supernovae ( SNe IIb ) , and find that extra mass removal by jets increases the parameter space for the formation of SNe IIb in this channel . To explore the role of eccentricity and the extra mass removal by jets in the GEE we use the stellar evolutionary code MESA binary . The initial primary and secondary masses are M _ { 1 ,i } = 15 M _ { \odot } and M _ { 2 ,i } = 2.5 M _ { \odot } , respectively . We examine initial semi-major axes of 600 - 1000 R _ { \odot } , and eccentricities in the range of e = 0 to e = 0.9 . Both Roche lobe overflow ( RLOF ) and mass removal by jets , followed by a wind from the SN IIb progenitor , leave a hydrogen mass in the exploding star of M _ { H,f } \approx 0.05 M _ { \odot } . This is compatible with a SN IIb progenitor . We find that in many cases , with and without the extra mass removal by jets , the system can enter a common envelope evolution ( CEE ) phase , and then gets out from it . Despite the large uncertainties , extra mass removal by jets substantially increases the likelihood of the system to get out from a CEE . This strengthens earlier conclusions for circular orbits . In some cases RLOF alone , without mass removal by jets , can form SN IIb progenitors . We estimate that the extra mass removal by jets in the GEE channel increases the number of progenitors relative to that by RLOF alone by about a factor of two .