Ramanujan-type congruences for the Andrews spt(n) partition function have been found for prime moduli 5 ≤ ℓ ≤ 37 in the work of Andrews [Andrews GE, (2008) J Reine Angew Math 624:133-142] and Garvan [Garvan F, (2010) Int J Number Theory 6:1-29]. We exhibit unexpectedly simple congruences for all ℓ≥5. Confirming a conjecture of Garvan, we show that if ℓ≥5 is prime and (-δ/ℓ) = 1, then spt[(ℓ2(ℓn+δ)+1)/24] ≡ 0 (mod ℓ). This congruence gives (ℓ - 1)/2 arithmetic progressions modulo ℓ(3) which support a mod ℓ congruence. This result follows from the surprising fact that the reduction of a certain mock theta function modulo ℓ, for every ℓ≥5, is an eigenform of the Hecke operator T(ℓ(2)).