Going through hiring now, the best fellowship is a damn communication fellowship. One that results in you having references who sing your praises, not just say "Eh, he was reasonably competent but I don't remember him that well". So many applicants can't string five words together on a CV, or interview and say dumb things, or refuse to do certain things, or have a track record of bouncing around due to poor behavior. It goes on like this.
If you have references who sing your praises and can say why, it speaks volumes. Our last few hires have ALL had this without fail. Most of our recent rejections have not.