Having meter 10.6.10.6 with refrain, it is always set to the tune which Root originally composed for it, which does not have a separate name.
There are many songs with similar themes and titles - this is one of the few which is definitely in the public domain, but nevertheless an on-line recording of it has not been found: please leave a comment below if you know of one.
LyricsShe only touched the hem of His garment
As to His side she stole,
Amid the crowd that gathered around Him,
And straight away she was whole.
I’ve touched the hem of His garment,
And now I, too, am free;
His healing pow’r this very hour,
Gives life and health to me.
She came with fear and trembling before Him—
She knew her Lord had come;
She felt that from Him virtue had healed her,
The mighty deed was done.
He turned with, “Daughter, be of good comfort,
Thy faith hath made thee whole,”
And peace which passeth all understanding
With gladness filled her soul.