I'd do
{
command1
command2
} | tee logfile-1.txt
{
command3
command4
} | tee logfile-2.txt
Alternatively, backup the original filedescriptor, first:
exec 6<&1
exec > >(tee log1)
echo 1
echo 2
exec 1<&6 6<&-
exec > >(tee log2)
echo 3
echo 4
See http://www.linuxtopia.org/online_books/advanced_bash_scripting_guide/x13082.html for many more advanced uses of exec