Your options for an improved bike I would say are rebuild what you have, look for something used (I have had great luck with CL and garage sales), something like a Wal-Mart bike (not even a choice in my mind unless you look on line and I think they may have a couple there better quality), on line new (I have a bikes direct bike I’m quite happy with), sporting goods store bike (step above Dept store bike IMO) and lastly a bike shop.
So much of the choice has to do with your abilities. If you want to have any extensive amount of work done by a shop you will be ahead to buy new. I have found many high quality bikes used that needed nothing or minimal work to get them back to new condition. But you have to know what you are looking for and at. If your service skills are not the best and you are looking on line see if you can get a bike shop to work with you before you order. Sporting goods stores as well as bike shops may or may not overlap both in quality of bikes and skill of employees. In general an owner operator bike shop that sells a high end bike even if what you are buying is mid-range will have quality people to help.
The only cost savings with something on line over a shop bike is the prep work the shop does and the cost of the shop to carry the inventory. The plus is with a shop you can actually get fitted and ride the bike before you buy it. For many that’s worth the $100 or so you might be looking at and being able to take it back for checkups.