I found a way to do it without updating initramfs.
I only added a single line to /etc/modules.
# /etc/modules: kernel modules to load at boot time.
#
# This file contains the names of kernel modules that should be loaded
# at boot time, one per line. Lines beginning with "#" are ignored.
ahci
The module libahci will be automatically loaded because the module ahci depends on it.
After rebooting the machine I entered UEFI-settings and switched the SATA-mode to AHCI. Now the system boots in AHCI-mode. There was no need to run update-grub or reinstall Grub.
Some additional information
I am using UbuntuStudio 16.04 (xfce). Enabling only AHCI-mode in BIOS/UEFI-settings led me to Grub's command line interface and the system wouldn't boot. I also read a lot of threads about the topic, many of them said it should work out of the box. This seems to be true for Ubuntu (which I had installed previously in IDE-mode too), but obviously not for Xubuntu or UbuntuStudio although I couldn't find any differences in the kernel configuration files. So it seems to depend on the distribution we use. One obvious difference during my tests were the used kernel versions (4.4 low-latency on UbuntuStudio 16.04 and 4.10 HWE on Ubuntu 16.04), so that might be a reason.
I'm using a 1 TB HDD with 64 MB cache. Checking read speed with
mook@MookPC:~$ sudo hdparm -Tt /dev/sdb
[sudo] password for mook:
/dev/sdb:
Timing cached reads: 10094 MB in 2.00 seconds = 5048.93 MB/sec
Timing buffered disk reads: 550 MB in 3.00 seconds = 183.16 MB/sec
give me almost the same values in IDE-mode or AHCI-mode, but I experienced a slight faster boot-time (25s in IDE-mode, 22s in AHCI-mode) and opening a folder with a lot of images in Phototonic seems to work noticeably faster with AHCI.
I also think that AHCI-mode will reduce the wear of moving parts in the HDD.