If you look at the design of a Shimano FD, you will find a channel in the middle
Your FD will be similar to this one
If you review the Shimano triple FD specification, then you will find "Applicable Top-Mid tooth difference ", which for MTB type (FC-M) is typically 10t, and for trekking type (FC-T) typically 12t.
In your case you have 28-38-44t, which is a 6t gap.
This will not work well because the channel is positioned for a middle ring 12t smaller than the upper ring.
In the case of FC-T4010, there are two offerings - 48-36-26, or 44-32-22. A derailleur such as FD-T4000 which you might have fitted, is specced to work perfectly with both - they are essentially the same thing, just 4t further down.
A 6t gap is not in spec at all. Here it seems like you might do well switching to 48-36-26, but that would mean three new chainrings.
Note that the chainrings are sold in sets, in that the 36t chainring is designed with ramps & pins which will pick up from a 26t ring.
In the past Shimano made 8-speed 48-38-28 chainsets such as FC-M361, which you could also use, or switch the chainrings from those onto your bike. But you won't get 44-38 to work well.
Note also that if you have an 8-speed drivetrain and a 9-speed chainset (which T4010 is) then the 9-speed chain is narrower, and the FD cage would be narrower also. So depending on the FD that you have (8 or 9 speed), you would want to use:
- 9-speed FD with 9-speed chainrings and 9-speed chain and 8-speed cassette (RDs have no 'speeds')
- 8-speed FD with 8-speed chainrings and 9 or 8-speed chain and 8-speed cassette
It's not clear which chainrings you have purchased, but if you purchase a 48t chainring for a FC-M361 chainset, then it will pickup well from a 38t chainring, however these are 8-speed chainrings which are possibly not ideal if you otherwise have 9-speed chainrings fitted. However you'd want to replace the FD for a suitable '10t top-mid' type, such as FD-M313 (which is 8-speed cage)
Instead of replacing with a 10t-type FD, you could also fit a whole new chainset; in this case note that you have an Octalink BB (which is essentially an obsolete system Shimano introduced to save weight over square taper, and replaced with HT2), and e.g. T4060 is HT2 (new BB required), and T3010 is square taper (BB also required, but good-quality cheap square taper BBs are going to become harder to find, so this is not ideal). If your BB is reaching the end of its life than you might go for HT2 as a replacement; if it's not, FC-T4010 48-36-26 would be a straight swap with no BB change needed.