We present an improved design for electrically pumped vertical external-cavity surface-emitting lasers (EP-VECSELs) optimized for passive modelocking with a semiconductor saturable absorber mirror (SESAM). In continuous-wave (cw) multimode operation, up to 170 mW of output power are demonstrated at a wavelength of 967 nm. Overcoming our prior limitation, we demonstrate fundamental transverse mode operation with 26 mW of average output power. Passively modelocking one of the fabricated EP-VECSELs with a quantum-well SESAM, we have achieved 6.3-ps pulses with an average output power of 6.2 mW at 1.46-GHz repetition rate, the shortest pulse duration from this type of laser to date. With slightly longer pulses of 7.3 ps, an average output power of 13 mW is obtained. Tradeoffs for further output power scaling and reduced pulse duration are discussed.