In all honesty 56cm sounds like a small frame size for someone who is 6' 1'' for some context I am 5' 8'' and generally find 56cm to be my preferred fit. All else being equal a 58 cm will likely give you more fitting options, as the frame stack will be taller. Most newer riders eventually complain of not being able to get the bars high enough, few complain they can't get them low enough (unless they are aiming for a particular aesthetic). (Your reach often lengths with higher bars so you don't have to fear a larger frame.)
By swapping parts (e.g., stem, handle bars) and making adjusting it should be possible to get one frame size to fit like the other. The real difference between the two sizes is how extreme can you go. The 56 cm will be limited in how high (stack) you can get while the 58 cm will likely be limited in far you can shorten the reach. Also you can try even larger (e.g., 60 cm) as a reference point.
It can often take many hours in the saddle before realizing one is a little too small or large (e.g., took me 1/2 year to realize one frame was too small causing me back issues). The best end goal I have found is finding a fit that leaves you balance - that is someone could quickly make your handle bars disappear you wouldn't fall forward while pedalling. The balance point will differ by some of your basic body dimensions and how aggressively you ride on average.
Modern obsession with smaller frames
For some reason many bike shops are obsessed with getting people on smaller road frames, but arguments can be made for sizing up (called a French Fit). If you plan on doing more endurance riding (or relaxed riding) then a larger frame can be easier to get into a comfortable position (i.e., taller stack and more reach).
Smaller frames are generally considered for "performance" fits and tend to be pushed by many road shops these day. You combine a smaller frame with a longer stem to make up for the shorter reach. The idea is to get low and long. This is how the "pros" fit their bikes, and it is really intended for someone who is riding at a very hard effort all the time.
When you ride at a hard effort your core muscles are constantly engaged and the force of pedalling supports most of your upper body weigh. Often you will need to get your upper body as low as possible to provide sufficient resistance to the pedal force as well as improving aerodynamics. As a result you will have little weight on your hands and you can get away with a lot of bad ergonomics in terms of how far you are reaching.
In a race or a hard club ride I find this type of fit is great, and I typically find a smaller bike with a long stem (e.g., 54 cm + 110 or 120mm stem) works great for me. However, the moment I want to go on a slower ride with the kids or a friend I find this type of fit unbearable. When I was younger I could suck it up, but not so much anymore.
Most do not want to ride at max effort everywhere. For an endurance fit (i.e., everyday riding), you want to be more upright and a larger frame. You are generating less pedalling force so you will not be able to support your upper body with your pedal force if you are leaned too far forward. A larger frame makes it easier to get your upper body in a more upright position (the stack will be taller so you have vertical height to work with) If you are more upright your reach will often length so the slightly longer reach of a larger frame will also work well here.
If you need to go faster you will often use your drops more in an endurance fit. In a performance fit, many find the drops too low to use except in moments of all out sprints or pursuits. In contrast in an endurance fit I find myself in my drops about 50% of the time while riding a constant pace. Sometimes the larger frames can feel funny at first, but I find it doesn't take long before you get used to it and I think part of this is the result of too many people are on bikes that are too small chasing a "pro" look.
- Take some time to consider the type of riding you will likely be doing most the time:
- If it is more competitive, then the smaller size may work for you.
- If it is more casual, then consider a larger frame.
- Go back and try the frames again, also try getting the staff to flip the stems up or down or move spacers around so you can try a few different positions.
- Finally, don't neglect the saddle position as this can totally change the feeling of a fit if the saddle is not set up properly (e.g., angled too up or down, set backward/forward too much).