The really good news is that reflux is almost standard with new babies and mostly sorts itself out with a bit of time.
Keeping baby upright as much as possible and for a good 10-20 minutes after feeds can help tremendously. DS had silent reflux but our pediatrician & gp both advised against medication unless it was very serious. I think they also suggested that staying upright would probably be more effective anyway.
I remember having to sleep sitting upright holding him every night for a few weeks early on - even putting the bassinet on an incline just wouldn't cut it. But then it just gradually got better and disappeared. Hopefully this will be a short-lived phase for you too.