There are plenty of flights with Award Availability on those dates:
[KVS Availability Tool 7.3.0/Diamond - Awards/OneWorld/AU-QFA-SP1]
Code:
NYC New York Metro NY US = JFK LGA
TFS Tenerife Sur ES [GCTS]
WED 23 Apr 2014 | 1 Seat
Carrier Flight From Depart To Arrive St Availability
--------- ------ ---- -------- ---- -------- --- -- -------------
AB 7451 JFK 17:15 DUS 06:55 +1 0 FS- CS- YS+
-> AB 2050 DUS 08:30 +1 TFS 12:15 +1 0 FS- CS- YS+
IB 6250 JFK 17:05 MAD 06:20 +1 0 FS- CS- YS+
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
AA 178 JFK 12:40 BOS 13:45 0 FS+ CS- YS-
-> IB 6166 BOS 17:30 MAD 06:20 +1 0 FS- CS- YS+
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
AA 323 LGA 11:15 ORD 13:00 0 FS+ CS- YS+
-> IB 6274 ORD 16:35 MAD 07:40 +1 0 FS- CS+ YS-
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
AA 327 LGA 11:40 ORD 13:25 0 FS+ CS- YS+
-> IB 6274 ORD 16:35 MAD 07:40 +1 0 FS- CS+ YS-
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
AA 331 LGA 12:45 ORD 14:25 0 FS+ CS- YS+
-> IB 6274 ORD 16:35 MAD 07:40 +1 0 FS- CS+ YS-
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
AA 335 LGA 13:35 ORD 15:15 0 FS+ CS- YS+
-> IB 6274 ORD 16:35 MAD 07:40 +1 0 FS- CS+ YS-
-> IB/I2 3910 MAD 11:55 +1 TFS 13:55 +1 0 FS- CS+ YS-
Sorry I meant in economy so YS+. You will notice that only one flight here has YS+ fully available - the others will get filtered out due to one or more legs not being available on YS. The AA website shows many economy low-redemption options.