Billboard had 4 charts from 1955 through 1957 and 3 charts up until the Hot 100 made it's debut in late 1958. I will list the highest position that any single reached among those 4 charts from 1956 through August 1, 1958.
Heartbreak Hotel > #1 > 1956
I Was The One > #19 > 1956
Blue Suede Shoes > #20 > 1956 > From an EP
I Want You, I Need You, I Love You > #1 > 1956
My Baby Left Me > #31 > 1956
Hound Dog > #1 > 1956
Don't Be Cruel > #1 > 1956
Love Me Tender > #1 > 1956
Anyway You Want Me > #20 > 1956
Love Me > # > 1956 > From an EP
When My Blue Moon Turns To Gold Again > #19 > 1956 > From an EP
Poor Boy > #24 > 1956 > From an EP
Too Much > #1 > 1957
Playing For Keeps > #21 > 1957
All Shook Up > #1 > 1957
Peace In The Valley > #25 > 1957 > From an EP
(Let Me Be Your) Teddy Bear > #1 > 1957
Loving You > #20 > 1957
Jailhouse Rock > #1 > 1957
Treat Me Nice > #18 > 1957
Don't > #1 > 1958
I Beg Of You > #8 > 1958
Wear My Ring Around Your Neck > #2 > 1958
Doncha' Think It's Time > #15 > 1958
Hard Headed Woman > #1 > 1958
Don't Ask Me Why > #25 > 1958
One Night > #4 > 1958
I Got Stung > #8 > 1958
(Now and Then There's) A Fool Such as I > #2 > 1959
I Need Your Love Tonight > #4 > 1959
A Big Hunk o' Love > #1 > 1959
My Wish Came True > #12 > 1959
Stuck on You > #1 > 1960
Fame and Fortune > #17 > 1960
It's Now or Never > #1 > 1960
A Mess of Blues > #32 > 1960
Are You Lonesome Tonight? > #1 > 1960
I Gotta Know > #20 > 1960
Surrender > #1 > 1961
Lonely Man > #32 > 1961
Flaming Star > #14 > 1961 > From an EP
I Feel so Bad > #5 > 1961
Wild in the Country > #26 > 1961
(Marie's the Name) His Latest Flame > #4 > 1961
Little Sister > #5 > 1961
Can't Help Falling in Love > #2 > 1961
Rock-a-Hula Baby > #23 > 1961
Good Luck Charm > #1 > 1962
Anything That's Part of You > #31 > 1962
Follow That Dream > #15 > 1962 > From an EP
She's Not You > #5 > 1962
King of the Whole Wide World > #30 > 1962 > From an EP
Return To Sender > #2 > 1962
One Broken Heart for Sale > #11 > 1963
(You're the) Devil in Disguise > #3 > 1963
Bossa Nova Baby > #8 > 1963
Witchcraft > #32 > 1963
Kissin' Cousins > #12 > 1964
It Hurts Me > #29 > 1964
Kiss Me Quick > #34 > 1964
What'd I Say > #21 > 1964
Viva Las Vegas > #29 > 1964
Such a Night > #16 > 1964
Ask Me > #12 > 1964
Ain't That Loving You Baby > #16 > 1964
Do the Clam > #21 > 1965
Crying in the Chapel > #3 > 1965
(Such an) Easy Question > #11 > 1965
I'm Yours > #11 > 1965
Puppet on a String > #14 > 1965
Tell Me Why > #33 > 1966
Frankie and Johnny > #25 > 1966
Love Letters > #19 > 1966
Spinout > #40 > 1966
Indescribably Blue > #33 > 1967
Big Boss Man > #38 > 1967
U.S. Male > #28 > 1968
If I Can Dream > #12 > 1968
Memories > #35 > 1969
In the Ghetto > #3 > 1969
Clean up Your Own Back Yard > #35 > 1969
Suspicious Minds > #1 > 1969
Don't Cry Daddy > #6 > 1969
Kentucky Rain > #16 > 1970
The Wonder of You > #9 > 1970
I've Lost You > #32 > 1970
You Don't Have to Say You Love Me > #11 > 1970
I Really Don't Want to Know > #21 > 1970
Where Did They Go, Lord > #33 > 1971
I'm Leavin > #36 > 1971
Until It's Time for You to Go > #40 > 1972
Burning Love > #2 > 1972
Separate Ways > #20 > 1972
Steamroller Blues > #17 > 1973
I've Got a Thing About You Baby > #39 > 1974
If You Talk in Your Sleep > #17 > 1974
Promised Land > #14 > 1974
My Boy > #20 > 1975
T-R-O-U-B-L-E > #35 > 1975
Hurt > #28 > 1976
Moody Blue > #31 > 1976
Way Down > #18 > 1977
My Way > #22 > 1977
Guitar Man > #28 > 1981