diff --git a/pyfstat/core.py b/pyfstat/core.py
index e781c8828655000e9bd8a1aa0ee9e5515139e9f5..cdb23de115bd74932001e73c190567f0c5adc497 100755
--- a/pyfstat/core.py
+++ b/pyfstat/core.py
@@ -332,6 +332,7 @@ class ComputeFstat(BaseSearchClass):
     def __init__(self, tref, sftfilepattern=None, minStartTime=None,
                  maxStartTime=None, binary=False, BSGL=False,
                  transientWindowType=None, t0Band=None, tauBand=None,
+                 tauMin=None,
                  dt0=None, dtau=None,
                  detectors=None, minCoverFreq=None, maxCoverFreq=None,
                  injectSources=None, injectSqrtSX=None, assumeSqrtSX=None,
@@ -359,9 +360,11 @@ class ComputeFstat(BaseSearchClass):
             function, but with the full range, for debugging)
         t0Band, tauBand: int
             if >0, search t0 in (minStartTime,minStartTime+t0Band)
-                   and tau in (2*Tsft,2*Tsft+tauBand).
+                   and tau in (tauMin,2*Tsft+tauBand).
             if =0, only compute CW Fstat with t0=minStartTime,
                    tau=maxStartTime-minStartTime.
+        tauMin: int
+            defaults to 2*Tsft
         dt0, dtau: int
             grid resolutions in transient start-time and duration,
             both default to Tsft
@@ -659,6 +662,14 @@ class ComputeFstat(BaseSearchClass):
                     self.windowRange.tauBand = self.tauBand
                     if self.dtau:
                         self.windowRange.dtau = self.dtau
+                    if self.tauMin is None:
+                        self.windowRange.tau = int(2*self.Tsft)
+                    else:
+                        if not isinstance(self.tauMin, int):
+                            logging.warn('Casting non-integer tauMin={} to int...'
+                                         .format(self.tauMin))
+                            self.tauMin = int(self.tauMin)
+                        self.windowRange.tau = self.tauMin
 
             logging.info('Initialising transient FstatMap features...')
             self.tCWFstatMapFeatures, self.gpu_context = (
@@ -702,10 +713,6 @@ class ComputeFstat(BaseSearchClass):
             # true single-template search also in transient params:
             # actual (t0,tau) window was set with tstart, tend before
             self.windowRange.tau = int(tend - tstart)  # TYPE UINT4
-        else:
-            # grid search: start at minimum tau required for nondegenerate
-            # F-stat computation
-            self.windowRange.tau = int(2*self.Tsft)
 
         self.FstatMap = tcw.call_compute_transient_fstat_map(
             self.tCWFstatMapVersion, self.tCWFstatMapFeatures,
diff --git a/pyfstat/grid_based_searches.py b/pyfstat/grid_based_searches.py
index c7ef34630c1935914634e183be35939e3363ba6a..3a2a4cdd6e9a62ba89a010bb18e1154e7440609e 100644
--- a/pyfstat/grid_based_searches.py
+++ b/pyfstat/grid_based_searches.py
@@ -369,6 +369,7 @@ class TransientGridSearch(GridSearch):
                  detectors=None, SSBprec=None, injectSources=None,
                  input_arrays=False, assumeSqrtSX=None,
                  transientWindowType=None, t0Band=None, tauBand=None,
+                 tauMin = None,
                  dt0=None, dtau=None,
                  outputTransientFstatMap=False,
                  outputAtoms=False,
@@ -396,9 +397,11 @@ class TransientGridSearch(GridSearch):
             debugging). Currently only supported for nsegs=1.
         t0Band, tauBand: int
             if >0, search t0 in (minStartTime,minStartTime+t0Band)
-                   and tau in (2*Tsft,2*Tsft+tauBand).
+                   and tau in (tauMin,2*Tsft+tauBand).
             if =0, only compute CW Fstat with t0=minStartTime,
                    tau=maxStartTime-minStartTime.
+        tauMin: int
+            defaults to 2*Tsft
         dt0, dtau: int
             grid resolutions in transient start-time and duration,
             both default to Tsft
@@ -431,6 +434,7 @@ class TransientGridSearch(GridSearch):
             detectors=self.detectors,
             transientWindowType=self.transientWindowType,
             t0Band=self.t0Band, tauBand=self.tauBand,
+            tauMin=self.tauMin,
             dt0=self.dt0, dtau=self.dtau,
             minStartTime=self.minStartTime, maxStartTime=self.maxStartTime,
             BSGL=self.BSGL, SSBprec=self.SSBprec,
@@ -840,6 +844,7 @@ class FrequencySlidingWindow(GridSearch):
         self.nsegs = 1
         self.t0Band = None
         self.tauBand = None
+        self.tauMin = None
 
         if os.path.isdir(outdir) is False:
             os.mkdir(outdir)
@@ -957,6 +962,7 @@ class EarthTest(GridSearch):
         self.transientWindowType = None
         self.t0Band = None
         self.tauBand = None
+        self.tauMin = None
 
         if os.path.isdir(outdir) is False:
             os.mkdir(outdir)