You can find plenty of examples by considering surfaces isogenous to a product, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.

For an explicit situation, you can look at Corollary 2.5 of my paper

On surfaces of general type with $p_g=q=1$ isogenous to a product of curves, Communications in Algebra 36 (2008), no. 6, 2023-2053, arXiv:math/0601063.

You can find plenty of examples with multiple Albanese fibres by considering surfaces isogenous to a product, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.

For an explicit situation, you can look at Corollary 2.5 of my paper

On surfaces of general type with $p_g=q=1$ isogenous to a product of curves, Communications in Algebra 36 (2008), no. 6, 2023-2053, arXiv:math/0601063.

You can find plenty of examples by considering surfaces isogenous to a product, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.

For an explicit situation, you can look at my paper "On surfaces of general type with $p_g=q=1$ isogenous to a product of curves", Corollary 2.5.

You can find plenty of examples by considering surfaces isogenous to a product, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.

For an explicit situation, you can look at Corollary 2.5 of my paper

On surfaces of general type with $p_g=q=1$ isogenous to a product of curves, Communications in Algebra 36 (2008), no. 6, 2023-2053, arXiv:math/0601063.