Download the .tar.gz file for the version of make that you want to compile (install) from source from https://ftp.gnu.org/gnu/glibc/. For example, let's say I download make-4.3.tar.gz. wget can be used.

wget http://ftp.gnu.org/gnu/make/make-4.4.tar.gz

Use the tar extract command to extract the TAR file to the /tmp directory.

tar -zxpf make-4.4.tar.gz --directory /tmp/

Move into the extracted directory.

cd /tmp/make-4.4/

Run the configure command to configure make to be installed at /usr/local.

./configure --prefix=/usr/local

Run make (assuming you have some other version of make installed on your Linux system, which you should, as make is a default, built-in CLI).

sudo make

Run make install.

make install

This should install make in the /usr/local/bin directory.

]# /usr/local/bin/make --version
GNU Make 4.4
Built for x86_64-pc-linux-gnu
Copyright (C) 1988-2022 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <https://gnu.org/licenses/gpl.html>
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.

And re-source our shell so that the make CLI is in our $PATH.

source ~/.bashrc

And version the make command returns the installed version of make.

~]$ make --version
GNU Make 4.4
Built for x86_64-pc-linux-gnu
Copyright (C) 1988-2022 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <https://gnu.org/licenses/gpl.html>
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.